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6
votes
0answers
351 views

What is the structure of a space of $\sigma$-algebras?

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
1
vote
2answers
308 views

Topology on Set of Prime Filters of a Distributive Lattice

Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open ...
1
vote
2answers
219 views

Relationship of Bousfield Classes of Morava K-theories

Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle ...
1
vote
3answers
327 views

Triangulations of lattice polygons

Let L be a 2-dimensional lattice and P- a lattice polygon. Suppose, it is triangulated into lattice tiangles. What are restrictions on their areas? For instance, can a lattice triangle of even area ...
2
votes
1answer
278 views

Name of a lattice-property

Assume that we have a complete lattice $(L,\leq)$. I would like to know whether the following property has a specific name and whether lattices with this property have been studied somewhere: For ...
2
votes
0answers
207 views

Hurwitz integers and $F_4$

The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
2
votes
0answers
247 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
1answer
726 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
2answers
551 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
1answer
284 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
0answers
652 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
1answer
123 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
0answers
97 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
1answer
280 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 ...
1
vote
0answers
161 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
1answer
506 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
1answer
325 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
1answer
234 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 ...
1
vote
0answers
131 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
2answers
209 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
0answers
197 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
3answers
433 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
1answer
418 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
0answers
136 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
0answers
236 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
0answers
186 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 ...
6
votes
2answers
499 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
1answer
207 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
2answers
470 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
1answer
637 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
1answer
413 views

Lorentzian characterization of genus

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973) $$ U \; = \; \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right),$$ ...
15
votes
3answers
1k 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
2answers
391 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
1answer
393 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
1answer
276 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
1answer
508 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
2answers
393 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 ...
14
votes
1answer
655 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
0answers
169 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
1answer
314 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
1answer
464 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
2answers
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
2answers
468 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
2answers
519 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
2answers
175 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
0answers
150 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
2answers
514 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
6answers
541 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
1answer
874 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
1answer
447 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 ...