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67 views

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

This post is a dual version for the Generalization of a theorem of Øystein Ore in which we have proved:
Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...

**1**

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**0**answers

73 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 : ...

**2**

votes

**1**answer

74 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**

votes

**1**answer

119 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 ...

**1**

vote

**0**answers

72 views

### Lattices in $PSL_2(\mathbb{R})$ [migrated]

Can a lattice in $PSL_2(\mathbb{R})$ have a normal abelian subgroup? It looks to me that it doesn't, but where can I read a proof?

**6**

votes

**1**answer

238 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

99 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

107 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

**2**answers

192 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

40 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 ...

**2**

votes

**0**answers

105 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?

**2**

votes

**2**answers

128 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 ...

**13**

votes

**3**answers

618 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, ...

**3**

votes

**2**answers

176 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

182 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

101 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?

**0**

votes

**2**answers

68 views

### Information needed to distinguish combinatorially isomorphic polytopes (up to affine equivalence)

I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help.
The title pretty much ...

**6**

votes

**1**answer

118 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 ...

**5**

votes

**0**answers

79 views

### Bound on the number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit ...

**12**

votes

**0**answers

204 views

### Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...

**1**

vote

**0**answers

112 views

### Dislocations,Disclinations Latices, Displacement fields and scaling

I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...

**0**

votes

**0**answers

56 views

### Sumset of parallel arithmetic progressions in Euclidean space

Let $A$ be a finite subset of $\mathbb R^n$ such that dim $A=n$ ($A$ is not contained in any lower dimensional hyperplane). It is a known result from Freiman that the size of the sumset $A+A$ is lower ...

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votes

**0**answers

54 views

### Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?

Question
Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = ...

**5**

votes

**1**answer

178 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 ...

**0**

votes

**1**answer

72 views

### Functional Encryption for Inner Product Predicates

I want to try to implement a functional encryption scheme proposed in http://eprint.iacr.org/2011/410. The first problem I faced with is a TrapGen algorithm. In the paper theorem 3.1 states that:
...

**8**

votes

**2**answers

765 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 ...

**1**

vote

**0**answers

139 views

### Standard name for a Monoid/Semigroup with $a+b \leq a, b$?

I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice?
For instance, for reals $a,b > 0$, define $$a \oplus b = ...

**4**

votes

**1**answer

115 views

### When are the congruence lattices nicer?

This is a purely idle question, but one I'm increasingly interested the more thought I put into it:
For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...

**6**

votes

**0**answers

125 views

### Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying
$b$ is bilinear,
...

**3**

votes

**0**answers

251 views

### What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
...

**2**

votes

**0**answers

71 views

### Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}^n$. Consider the real (positive) convex cone $\mathbb{R}_+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...

**5**

votes

**2**answers

157 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
...

**7**

votes

**1**answer

291 views

### Examples of fundamental domains

Everyone knows that it's difficult to compute a general fundamental domain for an arithmetic group but are there any specific examples where such domains have been calculated? I'm mostly interested in ...

**5**

votes

**1**answer

315 views

### Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$

For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$.
It is well known that in dimension ...

**8**

votes

**1**answer

204 views

### Why do the projections in the Calkin algebra not form a lattice?

Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and ...

**0**

votes

**0**answers

41 views

### Is it true that the set of points minimizing their distance to a multiset of intervals from a distributive lattice is an interval?

Let $(E, \preceq)$ be a finite distributive lattice, $H_E$ be the Hasse diagram of $E$ and $d$ be the distance on $E \times E$ defined as the length of the shortest path in $H_E$ between any pair of ...

**1**

vote

**0**answers

175 views

### A question on the poset of classes of isomorphic subgroups of finite groups

Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where
$[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining
...

**3**

votes

**0**answers

106 views

### Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows:
For a ...

**22**

votes

**0**answers

398 views

### 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 ...

**4**

votes

**1**answer

110 views

### Periodic functions over different lattices in $\mathbb R^d$ are linearly independent [closed]

I have the following claim that I think have been proved by someone, but I can not find the reference, hence I would like to ask for help. Here is the claim:
Let $f_1, \ldots, f_n$ be continuous ...

**2**

votes

**1**answer

139 views

### generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?

**0**

votes

**1**answer

84 views

### A limit of a sum related to integer lattice and power series

I have the following lemma that I would like to find a source to cite for. Let $L$ be a subset of $\mathbb Z^d_{>0}$. I would like to claim that the limit
$$\lim_{z \to (1,\ldots,1)^-} (\sum_{v ...

**1**

vote

**0**answers

60 views

### Is this related to a simple property of a lattice?

I am looking for a certain notion of sparseness of lattices.
I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I ...

**5**

votes

**4**answers

279 views

### Is every Heyting algebra the Lindenbaum algebra of an intuitionistic first order theory?

This question comes after the comments in the recent related question Sigma-complete Lindenbaum algebras?, but in its current form is sufficiently different in my opinion, and so I decided to follow ...

**1**

vote

**1**answer

196 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 ...

**0**

votes

**0**answers

99 views

### Krein-Rutman version of Hahn-Banach

Consider an arbitrary set of normed Riesz spaces $(X_i,\Vert \cdot \Vert_i,\leq)$, $i\in I$ ($I$ can be compact).
Can I apply the Krein-Rutman version (
see Schaefer; TVS; Corollary 2 of 5.4, ...

**3**

votes

**1**answer

169 views

### Is this bounded from below?

Let $u_1, u_2, u_3 \in \mathbb{Z}$ such that $u_1^2 + u_2^2 = u_3^2$.
Is $(u_3 + \frac{u_1 + u_2}{\sqrt{2}})^2$ bounded from below?
The irrationality of $\sqrt{2}$ certainly precludes zero, but can ...

**2**

votes

**0**answers

249 views

### 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 ...

**9**

votes

**0**answers

345 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, ...

**3**

votes

**1**answer

85 views

### existence of more distributive Boolean lattices

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in ...