**2**

votes

**0**answers

288 views

### Constructing the Stone Space of a Distributive Lattice

Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian logics"? ...

**9**

votes

**1**answer

775 views

### Reference request: Ehrhart's conjecture on the geometry of numbers

Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n ...

**9**

votes

**2**answers

577 views

### Ellipsoids and lattices: an enclosure problem.

$E \subset {\mathbb R}^2$ is an ellipse of area $1$ centered at the origin that contains no other point with integer coordinates. Is there a matrix $A \in SL(2,{\mathbb Z})$ such that the ellipse ...

**0**

votes

**1**answer

306 views

### Finding lattice with short basis-vectors containing given lattice

Hello
While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$ I am would like ...

**18**

votes

**0**answers

731 views

### Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...

**5**

votes

**1**answer

125 views

### Is the cardinality of occuring torsion subgroups in cofinite lattices in SL(2,R) bounded?

Let $\Gamma$ be a cofinite lattice in $PSL(2,\mathbb{R})$ with torsion subgroup $H$.
Is the a uniform bound on the cardinality of $H$?

**0**

votes

**0**answers

101 views

### Question regarding contiguous forms

I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...

**1**

vote

**1**answer

334 views

### Complete De Morgan algebra

Recall that an algebra $(A,\sim)$ is a De Morgan algebra if $A$ is a bounded distributive lattice and $\sim$ is a unary operation which satisfies:
${\sim} (x\vee y)={\sim} x\wedge {\sim} y$ and ...

**2**

votes

**1**answer

233 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 ...

**4**

votes

**1**answer

590 views

### Lovász $\delta$ condition for LLL Algorithm

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm
What is the importance of the $\delta$ parameter for LLL bases called Lovász condition?
...

**2**

votes

**1**answer

339 views

### Extending a complete lattice to get a “nice” Boolean lattice

Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which complement(a) = lub{b | b /\ a = bottom} = glb{b | b / a = ...

**2**

votes

**1**answer

247 views

### Is there existing terminology for this technical condition on semilattices?

Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of ...

**2**

votes

**0**answers

155 views

### bounds for the covering radius (or diameter of the Voronoi cell) of a lattice coming from a number field

Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by ...

**1**

vote

**2**answers

212 views

### About lattice $\pmod q$

For any matrix $A \in Z^{n\times m}$, Let $$\wedge_q(A)=\{ y\in Z^m:\exists s\in Z^n, s.t. y=A^ts (\mod q) \},$$ $$\wedge_q^\bot(A)=\{x\in Z^m: Ax=0 (\mod q)\}.$$ There is a result stating that ...

**2**

votes

**0**answers

210 views

### Simultaneous diophantine approximation with polynomial bound

For a given number $\alpha$ continued fractions expansion $(p_n, q_n)$ of $\alpha$ has the remarkable property that not only $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q_n^2}$, but the converse holds - ...

**1**

vote

**3**answers

504 views

### Genus and Spinor genus of a lattice

Hi, I'm looking for a motivation for the names genus and spinor genus of a lattice (and spinor norm of an isometry).
Is there any relation between the genus of a lattice and the genus of an algebraic ...

**5**

votes

**1**answer

422 views

### Finding Generators of O( Z^3,x^2 + xy + y^2 - z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:
\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 ...

**2**

votes

**0**answers

144 views

### Minimally 6-connected 3D discrete lines that are convex lattice sets

There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that ...

**2**

votes

**0**answers

259 views

### Pointwise bounds for Dirichlet kernel over truncated lattice

In 1 dimension, a "one-sided" Dirichlet kernel $D_N(x)=\sum_{k=0}^{k=N}e^{\frac{2\pi}{N}ikx}$ has its module sharply peaked around points corresponding, roughly, to the "dual lattice" ...

**1**

vote

**0**answers

198 views

### Upper bound on the number of shortest vectors in a lattice

Given a Lattice $L$ (e.g. by its Gram-Matrix or via a basis) I would like to know whether there is an upper bound on the number of shortest vectors in $L$. Available information on $L$ includes ...

**7**

votes

**2**answers

546 views

### Maximal number of edges and triangular cells for n points in a triangular lattice

Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...

**1**

vote

**1**answer

213 views

### Representing integer points inside a polytope using a unit hypercube

Let $D\subset\mathbb{Z}\times\mathbb{Z}$ such that $D$ is formed by the points bounded by a convex polytope. Let $f:D\to\mathbb{R}$ defined by $f(x,y)=ax+by+c$, where $a,b,c\in\mathbb{R}$ and ...

**5**

votes

**2**answers

546 views

### Lattices in SOL

Consider a semi-direct product $\mathbb{Z}^2\rtimes_A\mathbb{Z}$, where $A\in SL_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional solvable Lie ...

**8**

votes

**1**answer

691 views

### The Number of Short Vectors in a Lattice

Given a lattice $L = \bigoplus_{i=1}^{m} \mathbb{Z}v_i$ (the $v_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the ...

**4**

votes

**1**answer

464 views

### Lorentzian characterization of genus

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
...

**17**

votes

**3**answers

2k views

### A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one

It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb Z$ and is positive, is in a genus containing only one ...

**6**

votes

**2**answers

434 views

### Is the square of the covering radius of an integral lattice/quadratic form always rational?

This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to ...

**1**

vote

**1**answer

404 views

### Scott topology, but for graphs

Hi,
Would it be possible to define an order theoretic topology on graphs? I am thinking about the scott topology. There would be an associated continuous map on graphs.

**5**

votes

**1**answer

289 views

### Orthogonal Complements of Root Lattices in E_8

I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $ E_8 $ lattice. Is the orthogonal complement of $ L$ in $E_8$ unique up to ...

**0**

votes

**1**answer

563 views

### Neron-Severi Lattice of Elliptic K3

I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass ...

**0**

votes

**2**answers

505 views

### Lattice generators

I hope this question isn't trivial. Let $L$ be a lattice in $\mathbb{C}$ generated by two complex numbers $w_1,w_2$ which are linearly independent over $\mathbb{R}$. Let $\gamma\in\mathbb{C}$ be a ...

**15**

votes

**1**answer

727 views

### Codes, lattices, vertex operator algebras

At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention ...

**0**

votes

**0**answers

176 views

### Ordering labellings of a fixed poset.

Let $\{A_1,\ldots, A_m\}$ be a family of sets and $I=\{1, \ldots, m\}$. Assume for any $J\subset I$, $B_J=\bigcap_{i\in J}A_j$ satisfies $1\leq |B_J| \leq m-1$ as long as $|J|>1$.
We define a ...

**4**

votes

**1**answer

342 views

### Integrating the multinomial over a hypercube

I have come across an integral of the form
$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$
I have a solution that makes use of the partition function, but I ...

**2**

votes

**1**answer

476 views

### Even lattices and binary codes

I have a maybe simple question about even positive-definite lattices and lattices coming from binary codes. They seemed to be used in framed vertex operator algebras.
What is known about even ...

**12**

votes

**2**answers

1k views

### How is the Ising model an example of a lattice model as per Kontsevich?

In section 3.2 of Kontsevich's very interesting paper "Notes on motives in finite characteristic,", he gives an axiomatic definition of a "lattice model" attached to a Boltzmann datum (V_1,V_2,R), ...

**4**

votes

**2**answers

543 views

### diameter of Voronoi cell of the lattice ? What about R^n ? What about small n =2,3,4 ?What about random lattice ?

Consider a lattice in R^n.
Consider Voronoi cell of it.
What is known about diameter ? About the shape ? What are good references ?
As far as I understand they are not easy to compute.
May be in ...

**7**

votes

**2**answers

530 views

### What groups have a second maximal subgroup below exactly four maximal subgroups?

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal ...

**2**

votes

**2**answers

180 views

### how to find vertex of parallelotope closest to given point P in R^n ? (Or minimize quadratic form over {+-1}) Is it NP ?

Consider a parallelotope in R^n and some point "P" in R^n.
What algorithms (except of brute force) can be suggested to find the closest vertex of paralleloptope to "P" ?
Is it NP ?
Parallelotope ...

**1**

vote

**0**answers

159 views

### How to find closest point to restricted lattice on the plane ? ( m*h1 + n*h2, for 0<m,n<N)

Consider finite piece of lattice i.e. points of the form m*h1 + n*h2, for 0<m,n<N h1, h2 -some vectors. Consider some point "P" on the plane. How to find ...

**3**

votes

**2**answers

534 views

### Geometric interpretation of distributivity

I'm studying distributivity in the lattice theory, consequently, I'm after any ideas that
might help to develop some intuition.
In elementary school level algebra, distributivity property
...

**9**

votes

**6**answers

566 views

### Which finite nonabelian groups have long chains of subgroups as intervals in their subgroup lattice?

Given N, what is a finite non-abelian (and preferably non-solvable) group G in whose subgroup lattice Sub[G] there is an interval that is a chain of length at least N?
Since N can be arbitrarily ...

**4**

votes

**1**answer

1k views

### Closest vector problem (=nearest lattice point) is trivial for “reduced lattice” ?

Consider some lattice in R^n. Take some point "P" in R^n (which does not belong to this lattice in general). The problem is to find "nearest" lattice point. The problem is known NP-hard in general it ...

**9**

votes

**1**answer

509 views

### Niemeier lattices and theta functions

I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice ...

**3**

votes

**1**answer

270 views

### How “often” does LLL-reduction produce “optimal” result ? Is there condition (or informal understanding) on lattice that it LLL is optimal ?

Consider some lattice in R^4 (C^4) or C^8.
Famous "lattice reduction" procedures (like LLL latice reduction)
produces some "reduced basis". However in general there results are not "the best ...

**5**

votes

**2**answers

532 views

### Lattice reduction in R^3 (R^4) or what is fundamental domain for SL(3,Z) , (SL(4,Z)) ?

Consider a lattice in R^3.
Is the some "canonical" way or ways to choose basis in it ?
I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|.
Considering lattices with ...

**7**

votes

**2**answers

1k views

### How to find nearest lattice point to given point in R^n ? Is it NP ?

Consider some lattice in R^n.
Take some point "P" in R^n (which does not belong to this lattice in general).
What are the algorithms to find some nearest lattice point to "P" ?
"Nearest" - means in ...

**1**

vote

**2**answers

302 views

### compact elements and continuous functors

Hi,
I am interested in abstracting the Scott topology from Domains to Categories. One can find a definition of a continuous functor which is just such an abstraction:
A functor F:C→D is continuous ...

**6**

votes

**2**answers

474 views

### Dense sphere packings which are not lattice packings

This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...

**0**

votes

**0**answers

135 views

### Bogus(?) proof for equivalence of normal and dual distributivity conditions in lattices

I am attempting to prove the equivalence of the following two definitions of distributive lattices:
$(a \lor b) \land c = (a \land c) \lor (b \land c)$
$(a \land b) \lor c = (a \lor c) \land (b \lor ...