**6**

votes

**0**answers

121 views

### 2-dimensional sublattices with all vectors having very big square (in absolute value)

QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not
definite, not necessarily unimodular, $n>2$. I want ...

**54**

votes

**5**answers

3k views

### How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...

**5**

votes

**1**answer

236 views

### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...

**0**

votes

**2**answers

241 views

### Intersection of two lattices

Suppose that $\Lambda_1, \Lambda_2$ are two sub-lattices of $\mathbb{Z}^n$ of full rank, defined by congruence modulo a prime $p$. That is, there exist two vectors with integer entries $\mathbf{a}, ...

**0**

votes

**0**answers

51 views

### lattice basis reduction of the orbit of a rational vector on the torus

LEt $v=(p_1/q,...,p_n/q)$ be a vector of the torus $\mathbb{T}^n$, such that for any $i$, $p_i$ and $q$ are relatively prime. Let $L= \{ kv \mod \mathbb{T}^n , k=0,...,q-1 \}$.
What is the lattice ...

**6**

votes

**3**answers

257 views

### Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...

**2**

votes

**2**answers

168 views

### Permutation covering of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...

**6**

votes

**1**answer

148 views

### Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?

Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$)
Or is it known that it cannot be such a lattice ?

**4**

votes

**1**answer

402 views

### A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups.
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: $\mathcal{L}(H\subset G)$ ...

**5**

votes

**2**answers

202 views

### Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community.
Setup: Let ...

**4**

votes

**1**answer

121 views

### Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have
\begin{equation*}
f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′).
\end{equation*}
Suppose $f$ and $g$ are supermodular, ...

**0**

votes

**0**answers

70 views

### mean value estimate $ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = \log \frac{T}{2\pi } + (2\gamma - 1) + O(T^{\delta})$

I was browsing through the recent paper of Bourgain and Watt on mean value estimates of the Riemann zeta function on the critical line.
$$ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = ...

**9**

votes

**2**answers

990 views

### The Gauss circle problem on a hexagonal lattice

Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...

**10**

votes

**1**answer

533 views

### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

**5**

votes

**1**answer

177 views

### Bounding the number of lattice points inside an $n$-dimensional ellipsoid

I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse.
That is, given an $n$-dimensional ellipsoid ...

**4**

votes

**2**answers

221 views

### Counterexample on completely distributive lattices

I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets ...

**-3**

votes

**2**answers

85 views

### does a lattice have a minimal item [closed]

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite.
I think that ...

**2**

votes

**1**answer

42 views

### covering radius of a lattice from cyclotomic extension

Given a finite field extension $L$ of $\mathbb{Q}$ of dimension $n$, there is a natural way to embed it into $\mathbb{R}^n$ such that the image of its ring of integers $\mathcal{O}_L$ is a lattice. If ...

**5**

votes

**0**answers

198 views

### When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...

**1**

vote

**1**answer

226 views

### Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...

**10**

votes

**2**answers

383 views

### Inequalities for averaging over partially ordered sets

Let's start from a classical inequality:
If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then
$(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$.
It can be written also in ...

**10**

votes

**3**answers

410 views

### Is a distributive lattice planar iff it admits no B3 sublattice?

A finite lattice is planar if it admits a Hasse diagram which is a planar graph (we want the Hasse diagram to be represented in the plane so that the $y$-coordinate of each element respects the ...

**1**

vote

**1**answer

124 views

### Uniqueness of minimal completions of a partially ordered set

The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff
$X\subseteq Y$.
$\le'=(\le\cap X^2)$.
and a completion of $(X,\le')$ if in addition
$~~ 3$. ...

**8**

votes

**2**answers

944 views

### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

**4**

votes

**1**answer

122 views

### Can any finite distributive weighted lattice be realized by inclusion of groups?

By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice.
A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...

**2**

votes

**1**answer

242 views

### Implementation of the bounded-distance decoder of Leech-lattice?

Hi all,
I am wondering, anybody can help me how can I find an implemented version of Leech-Lattice quantizer/decoder, i.e., "Matlab", "C++" or "Python" code, using the approach proposed by Ofer ...

**33**

votes

**16**answers

3k views

### Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...

**0**

votes

**0**answers

50 views

### kernel lattice example

Could anyone give an example of the following?
Suppose that $A \in \mathbb{Z}^{m\times n}$, where $m \leq n^{1-\epsilon}$ for some $\epsilon > 0$. The entries of $A$ have size $poly(n)$, meaning ...

**3**

votes

**1**answer

177 views

### n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice.
Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...

**2**

votes

**1**answer

115 views

### lattice orthogonal complement

Let $A\in \mathbb{Z}^{m\times n}$ ($m<n$) be a matrix with orthogonal rows. Further assume that the gcd of the coefficients in each row of $A$ is $1$.
Consider $\ker A\cap \mathbb{Z}^n = ...

**1**

vote

**0**answers

82 views

### Are lattices in the special real linear group subgroup seperable?

Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : ...

**-1**

votes

**1**answer

179 views

### How to recognize if a lattice is distributive? [closed]

I know that a Boolean lattice must be distributive.
But what with these lattices? Are these distributive?
$\hskip0.7in$
How to recognize which lattices are distributive or not only by looking on ...

**12**

votes

**6**answers

2k views

### Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here.
Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$.
Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...

**6**

votes

**1**answer

297 views

### Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no ...

**1**

vote

**1**answer

132 views

### basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., ...

**0**

votes

**0**answers

191 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 ...

**2**

votes

**1**answer

515 views

### Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...

**2**

votes

**2**answers

235 views

### Is certain topology-related set a distributive lattice?

In my research the following problem appeared (and if it is true, this solves positively several my conjectures):
Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ($n$ ...

**0**

votes

**1**answer

63 views

### Lower bounds on the rank of a unimodular lattice, given the binlinear pairing of a subset of basis vectors

I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known.
Let $\Lambda$ be an odd, unimodular matrix of signature ...

**9**

votes

**4**answers

586 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 ...

**2**

votes

**0**answers

110 views

### $\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?

**3**

votes

**2**answers

147 views

### 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 ...

**18**

votes

**6**answers

730 views

### Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?

This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...

**4**

votes

**3**answers

494 views

### Product-Decomposition of distributive lattices

EDIT
I now (strongly) believe that the following claim answers my question (see the text below). However, if it does, then I am sure that it is known. It is not difficult to prove and the question ...

**6**

votes

**1**answer

270 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 ...

**13**

votes

**3**answers

624 views

### Conjecture regarding closest point inside a discrete ball to a line

I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, ...

**6**

votes

**1**answer

130 views

### Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...

**3**

votes

**2**answers

200 views

### Cohomology of SL(2,R) with coefficients given by linear action

Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.
What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$?
And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a ...

**5**

votes

**2**answers

211 views

### Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...

**3**

votes

**0**answers

113 views

### Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?

And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?