**1**

vote

**1**answer

221 views

### Commutative, idempotent partially ordered monoids

A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for commutativity). ...

**6**

votes

**3**answers

556 views

### Sums of inverse determinants over matrices

Let $A \in M_n(\mathbb Z)$ and $\|A\| = \max |a_{ij}|$.
Denote $$ S(r) = \sum_{\substack{\|A\| \leq r \\\ \det{A} \neq 0}} \dfrac{1}{|\det{A}|} $$
- the sum over all matrices $A \in M_n(\mathbb Z)$ ...

**6**

votes

**0**answers

154 views

### Minkowski's convex body theorem for ellipsoids

Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector.
Can this bound be improved ...

**2**

votes

**0**answers

167 views

### A question on the theorem of Minkowski-Hlawka

The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. Then there is a ...

**1**

vote

**1**answer

114 views

### General and translational Birkhoff lattices. Equational classes.

By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to be:
Is there an ...

**10**

votes

**4**answers

910 views

### 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 ...

**8**

votes

**6**answers

1k 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 ...

**8**

votes

**1**answer

193 views

### Is the group of integer points of ${\rm SO}(n,1)$ maximal?

That is, is it true that there does not exist a lattice in $G = {\rm SO}(n,1)$ which contains the group of integer points of $G$ as a proper subgroup (obviously then of finite index)? if such a ...

**0**

votes

**0**answers

127 views

### Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.

I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...

**1**

vote

**1**answer

305 views

### Minkowski's successive minima: A quantity not much larger than det(L)^(1/n) and not much smaller than λ_n(L)?

Let $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ be $n$ linearly independent vectors in an $n$-dimensional lattice $\Lambda$ in $\mathbf{R}^n$ and let $\mathbf{v}^*_1 ,\mathbf{v}^*_2, ..., ...

**1**

vote

**1**answer

126 views

### Lattice basis with Gram-Schmidt vectors of increasing length

Let $\Lambda$ be an $n$-dimensional lattice in $\mathbb R^n$ and let $\cal B$ be the set of all bases that generate $\Lambda$. For a basis
$\mathbf{B}=[\mathbf{b}_1, ... ,\mathbf{b}_n]\in {\cal B}$, ...

**5**

votes

**0**answers

205 views

### Lattice points inside a (n-dimensional) tetrahedron

Hi, overflowers.
I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperplanes and ...

**5**

votes

**2**answers

478 views

### Area of a lattice polygon in terms of its width

Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$).
Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w_v(M)=\max\limits_{x,y\in ...

**4**

votes

**1**answer

251 views

### A question of compactness in the geometry of numbers

Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the critical determinant of $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices ...

**8**

votes

**4**answers

487 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 ...

**5**

votes

**1**answer

150 views

### Lattice in motion group

Let $\Gamma$ be a discrete cocompact subgroup of the euclidean motion group
$$
G={\mathbb R}^d\rtimes O(d).
$$
Let $\phi:G\to O(d)$ the projection homomorphism.
Is it true that $\phi(\Gamma)$ is ...

**10**

votes

**2**answers

569 views

### discrete subgroups of Lie groups and actions on homogeneous spaces

Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on ...

**0**

votes

**1**answer

102 views

### orthogonality in a lattice

Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3.
...

**1**

vote

**1**answer

116 views

### orthogonal base in unimodular lattice

Let $\Lambda$ be an unimodular lattice with a quadratic form $(-,-)$ of signature $(m,n)$ , $m,n>0$.
I know that, fixed a base $e_1,\cdots,e_{m+n}$ for $\Lambda$, the matrix which has entries ...

**4**

votes

**2**answers

521 views

### Property of lattices in Lie groups

Let $\Gamma$ be a lattice in a (real or p-adic) Lie group.
Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each $\sigma\in\Sigma$ is ...

**2**

votes

**1**answer

119 views

### Distributive lattice embedding into a finite lattice.

Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice.
If $\iota$ preserves all meets and joins one can show that ...

**2**

votes

**2**answers

309 views

### Banach lattice subspace of $C([0,1])$ not a sublattice

This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice ...

**1**

vote

**1**answer

101 views

### distributive sublattices of atomistic ortholattices

Let $L$ be an atomistic ortholattice (i.e. every element can be written as a join of atoms) with top and bottom elements 0 and 1, and let $M$ be a distributive atomic sub-ortholattice of $L$.
Is $M$ ...

**6**

votes

**1**answer

416 views

### genus and spinor genus over a number field

Let $F$ be a number field with ring of integers $\mathfrak{o}$. Let $(V,Q)$ be a quadratic space of dimension $n$ over $F$, and let $L$ be a free lattice in $V$ (i.e. $L\cong\mathfrak{o}^n$). If the ...

**1**

vote

**0**answers

73 views

### Reference request for gluing construction of lattices

I would like to study gluing method of lattices (such as constructing Niemeier lattices from certain root lattices etc) and am looking for good references. I am aware of the book "Sphere Packings, ...

**9**

votes

**1**answer

309 views

### Lattice in a certain Lie group

Let $G_n$ be the Lie group consisting of $n \times n$ upper triangular matrices of determinant $1$ with real entries. In other words,
$$G_n = \{\text{$\left(\begin{matrix} a_{11} & a_{12} & ...

**1**

vote

**0**answers

183 views

### existence of order preserving map [closed]

suppose A is a linear order set with a copy of rationals in it that is $A=B\cup\{\bar{r}:r\in\mathbb{Q}\cap[0,1]\}$. is there an orde preserving map that preservs sup and inf betwwen A and $[0,1]$ in ...

**1**

vote

**1**answer

103 views

### Help with (Coxeter?) lattice identification.

I'm trying to find information about a specific lattice, which is proving difficult since I am not sure what its standard name is.
Consider the regular $n$-Simplex embedded in $\mathbb{R}^n$ with one ...

**1**

vote

**0**answers

84 views

### Information about mutant Leech lattice related to smallest perfect squared square

What happens if we follow the construction of the Leech lattice but replace the relation
$\displaystyle \sum_{n=1}^{24} n^2 = 70^2$
with the smallest perfect squared square? Explicitly, if we set ...

**2**

votes

**1**answer

287 views

### Probability that a “closable” self-avoiding random walk forms a polygon

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...

**5**

votes

**1**answer

219 views

### Complete anti-chain lattices and the axiom of choice

Hello, everyone. I'm trying to find out about lattices of anti-chains, and was wondering whether you could help me with getting to grips with a Comp. Sci. paper I'm struggling with.
I've been reading ...

**2**

votes

**4**answers

384 views

### Minimal (semi)lattice containing a given poset

For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I ...

**2**

votes

**2**answers

253 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**6**

votes

**2**answers

333 views

### Measuring how far from being cocompact a lattice is

Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by ...

**11**

votes

**1**answer

279 views

### Defining measures over frames in place of $\sigma$-algebras

Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...

**3**

votes

**1**answer

458 views

### Maximizing the number of lattice points in a circle of radius $r$ placed on a lattice

I have a circle of radius $r$, and I wish to place this circle of a $Z^2$ integer lattice or an $A_2$ hexagonal lattice s.t. I maximize the number of lattice points within or along the contour of the ...

**2**

votes

**2**answers

648 views

### An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

In a previous question: (The Gauss circle problem on a hexagonal lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a ...

**1**

vote

**0**answers

56 views

### variant on ring objects in the category of complete lattices

Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a ...

**4**

votes

**1**answer

221 views

### Most orthogonal lattice basis

Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...

**2**

votes

**1**answer

101 views

### Is the complete lattice of inflators on a frame a frame?

Yup, it may sound like an inocent question to many of you; but a very good friend of mine is completely baffled in his research about lattices of inflators on a frame. He asked me very kindly to post ...

**3**

votes

**0**answers

127 views

### references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN.
I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...

**4**

votes

**1**answer

279 views

### Invariant lattice of algebraic surface.

Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or ...

**1**

vote

**1**answer

139 views

### Covering a $d$-dimensional integer lattice by repeating a minimal set of deterministic moves

Imagine I place a turtle on some desired vertex, $v_i$, of a bounded $d$-dimensional integer lattice, $Z^d$, with dimensions $(l_1, ..., l_d)$. The turtle is able to travel from vertex to vertex ...

**1**

vote

**0**answers

123 views

### How many more join-irreducibles can there be in a sub join-semilattice of a finite lattice?

Let $L$ be a finite lattice. Then $L$ is generated by its join-irreducible elements $J(L)$ or alternatively its meet-irreducible elements $M(L)$.
If $S \subseteq L$ is a sub join-semilattice then ...

**7**

votes

**1**answer

192 views

### Restricting representations to lattices

Let $V$ be a finite-dimensional irreducible representation of the Lie group $\text{SL}_n(\mathbb{R})$. Must $V$ remain irreducible when you restrict the action to $\text{SL}_n(\mathbb{Z})$? More ...

**1**

vote

**1**answer

267 views

### The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself

What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...

**0**

votes

**0**answers

143 views

### $T^2$-fibered K3 surface with involution

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...

**2**

votes

**0**answers

266 views

### Singular fibers of an elliptic fibered K3 surface.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...

**5**

votes

**0**answers

692 views

### On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...

**3**

votes

**0**answers

157 views

### Automorphisms of Torsion Quadratic Forms

Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - ...