Questions tagged [lattices]
Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
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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 $\...
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coloring in lattice
This is a mathematical question raised from engineering and physics:
Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
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2
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Products of maximal inclusions of finite groups with a non-obvious intermediate
Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
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Are the homogeneous single chain subfactors, Dedekind?
Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
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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 ...
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Compactness and completeness in Gödel logic
The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
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in search of a transformation between determinants
Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A_n=\det B_n=2^{\...
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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 ...
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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 ...
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Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths?
The sequence A006318 at OEIS stands for the Schröder numbers.
They describes the number of lattice paths from the southwest corner $(0,0)$ of an $n\times n$ grid to the northeast corner $(n,n)$, ...
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Siegel's Mean Value Theorem by Rogers and Macbeath
I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that
$$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ ...
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Package for the Closest Vector Problem (CVP)?
Let $A$ be a positive definite, real $n \times n$ matrix. This defines a norm on $\mathbb{R}^n$. Now I have a given point $p \in \mathbb{R}^n$ and I want to find the lattice point $x \in \mathbb{Z}^n$ ...
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Automorphism groups of indefinite non-unimodular integer lattices
Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don'...
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Integrating an n-fold Cauchy product of a Fourier series
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
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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 ...
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A group-theoretic perspective on Frankl's union closed problem
Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
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Polar body of a convex body that avoids a lattice
Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K$...
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Understanding sphere packing in higher dimensions
In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved.
Admittedly it is very ...
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Is there an Ehrhart polynomial for Gaussian integers
Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
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Simple conjecture about rational orthogonal matrices and lattices
The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The ...
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Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
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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 to ...
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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 ...
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Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
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On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
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orbits of automorphism group for indefinite lattices
I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily ...
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Localic locales? Towards very pointless spaces by iterated internalization.
One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames,
certain sorts of ...
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24 vectors in Leech lattice having scalar product $\frac{1}{4}$ pairwise
Two vectors from Leech lattice - as defined on wikipedia - have scalar product $\pm 32,\pm 16, \pm 8$ or $0$. Do there exist 24 vectors having scalar product 8 pairwise ? When we consider unit vectors ...
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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, ...
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A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
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Chromatic number of Voronoi diagrams of lattices
Let $L$ be a Euclidean lattice. Define a graph whose vertex set is $L$ and where two points $x,y\in L$ are declared to be adjacent whenever the cells of $x$ and $y$ in the Voronoi diagram of $L$ have ...
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Best way to find a closest vector in a lattice
Let $v_1,\dotsc,v_n$ be linearly independent vectors in $\mathbb{R}^n$, and let $\Lambda=\bigoplus_{i=1}^n \mathbb{Z}v_i$. The question is, given a vector $w$ in $\mathbb R^n$, find the element $v$ ...
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
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An interesting sum over lattice points in a large disk centered at the origin
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D_r$ denotes the closed ...
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Which lattices have more than one minimal periodic coloring?
The lattice $\mathbb{Z}^n$ has an essentially unique (up to permutation) minimal periodic coloring for all $n$, namely the "checkerboard" 2-coloring. Here a coloring of a lattice $L$ is a coloring of ...
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Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24?
For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...
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Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$
Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of $\mathrm{SL}(2,\...
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Are there Type III codes with small but nonzero "index"?
Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...
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What kind of locally symmetric space is a rational sphere
Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...
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What's in the genus of the cubic lattice?
I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
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Counting points on the intersection of a box and a lattice
Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\...
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what is the number of paths returning to 0 on the hexagonal lattice
I am looking for an estimation of the number of paths of length $n$ going from 0 to 0 on the hexagonal (or honeycomb) lattice.
I can find plenty on references on self avoiding paths, but I am looking ...
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Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$
Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial?
The case where $G$ ...
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Extremal problem for 2-dimensional lattices
Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...
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Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
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Preserve validity between the two Kripke frames
The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame.
For $n \geq 1$, let $\mathcal{C}_n$ denote the frame which is shown in Fig.1. ...
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Is this property a new large cardinal notion?
Given a cardinal $\kappa$, $\kappa$-complete lattices are lattices that have joins and meets of less than $\kappa$ elements (in particular they are bounded). In what follows we shall restrict to the ...
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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 non-...
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Bounded version of linear and quadratic Hasse--Minkowski theorem
The Hasse-Minkowski theorem states that if
$$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$
is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation
$$Q(x_1,\...
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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 $R(L)= \{ x \in L : (x,x) =2\}$ ...