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1answer
59 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 $(X,\le)$?
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1answer
48 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 ...
0
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0answers
43 views

Number of lattice points inside a parallelogram defined by a vector on the “rhombic plane” [on hold]

I will give here a specific case and try to explain the question the best I can: The rhombic plane is the plane with the x-axis the same as the Cartesian plane but the y-axis tilted at $60$ degrees. ...
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0answers
50 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
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4answers
185 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
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0answers
128 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 ...
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0answers
95 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
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1answer
153 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
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0answers
212 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 ...
6
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0answers
152 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
1answer
60 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 ...
1
vote
1answer
274 views

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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5answers
493 views

Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?

This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...
4
votes
2answers
132 views

Bound on Minimal Length of Vectors in Lattice and its Dual Lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
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0answers
114 views

Is the direct product of distributive inclusions of groups, modular?

Let $H$ a subgroup of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups ($\mathcal{L}( G)$ if $H= \{ e \}$). Definitions: A lattice $(L, \wedge, \vee)$ is : - Distributive if ...
0
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2answers
118 views

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 ...
2
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0answers
166 views

Existence of inclusions of finite groups with a particular lattice property

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by : $(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
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0answers
86 views

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_{ ...
3
votes
1answer
58 views

Algebraicity of isogenies as maps of lattices

Let $E_i\colon y^2=4x^3+A_ix+B_i$, for $i=1,2$ be two elliptic curves where $A_i,B_i \in \mathbb C$ are algebraic over $\mathbb Q$. For $i=1,2$ let $\Lambda_i\subseteq \mathbb C$ be the unique lattice ...
5
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0answers
254 views

Is this obfuscation scheme unbreakable?

I've just come across this popular article about a breakthrough (which can be purchased here), published in Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium by a team of ...
1
vote
1answer
345 views

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 ...
5
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3answers
363 views

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 ...
2
votes
1answer
84 views

Has the single sorted case of formal concept analysis been investigated?

A formal context in formal concept analysis is a triple $K = (G, M, I)$ where $G$ is a set of objects, $M$ is a set of attributes and the binary relation $I \subset G \times M$ shows which objects ...
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0answers
82 views

Dedekind–MacNeille completion of ordered abelian monoids

It's known that the Dedekind–Macneille completion of an ordered Abelian group necessarily is not an ordered Abelian group (and it is an ordered Abelian monoid). I want to know that what happened ...
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0answers
63 views

approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...
4
votes
2answers
174 views

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 $R(L)= \{ x \in L : (x,x) =2\}$ ...
24
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2answers
842 views

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 ...
27
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14answers
2k views

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 ...
1
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1answer
66 views

Uniqueness of minimal completions of a partially ordered set

The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff $X\subseteq Y$. $\le'=(\le\cap X^2)$. and a completion of $(X,\le')$ if in addition $~~ 3$. ...
2
votes
1answer
162 views

Finite-index free subgroups in lattices and matrix rings

It is a theorem of Selberg that a lattice $\Gamma$ in a linear group has a torsion-free subgroup of finite index. Page 64 in 'Introduction to Arithmetic Groups' by Dave Morris asserts these can be ...
2
votes
2answers
98 views

Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html.) The same is true for lattices in $\mathbb{R}^n$, ...
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2answers
220 views

Inequalities for averaging over partially ordered sets

Let's start from a classical inequality: If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then $(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$. It can be written also in ...
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0answers
51 views

Finding special vectors generated by a matrix

Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix. Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
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6answers
889 views

Was lattice theory central to mid-20th century mathematics?

Four years ago, I read a book on the history of mathematics up to 1970 or so. It was very interesting up until the end. The last few chapters, though, were on lattices. The author claimed that ...
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0answers
126 views

Sampling a two-dimensional Gaussian distribution at points along an integer lattice

Please consider a two-dimensional Gaussian of the general form: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, where $C$ is the peak of the Gaussian, i.e. the point at which the ...
6
votes
1answer
96 views

How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...
12
votes
2answers
233 views

Lattice points and convex bodies

Given are two convex bodies $K, L \subset \mathbb{R}^n$ that contain the origin as an interior point. Assume the number of integer points contained in $\lambda K$ equals the number of integer points ...
6
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0answers
79 views

Minimum of the product of linear forms over a lattice

In Chapter [IX.1] of Siegel's Lectures on the Geometry of Numbers it is shown that if we have $n$ linear forms $y_{j}=\sum_{k=1}^{n}{a_{jk}x_{k}},\quad j=1,\ldots,n$, with the coefficient matrix ...
2
votes
1answer
184 views

Is any order bounded continuous linear functionals a difference of positive continuous functionals?

Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the ...
1
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0answers
63 views

Determining the position of a coordinate by binning Gaussian noise around that coordinate to lattice points with vertex-specific probabilities [closed]

(NOTE: I have changed and hopefully simplified this question by removing the section on randomly perturbing lattice points, and instead specifying that the counts at each vertex should be randomly ...
6
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1answer
231 views

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 ...
2
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1answer
90 views

Achieving the largest possible minimum spacing between vertices of the same color in an integer lattice

Consider an infinite integer lattice, or an infinite hexagonal lattice with unit length edges. Provided a set of $k$ possible vertex colors, is there a known largest possible minimum spacing that can ...
2
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0answers
79 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
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3answers
167 views

Crystal structure, lattice, Graph and coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field. Given a periodic graph (actually a physical ...
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3answers
352 views

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 ...
4
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2answers
386 views

Do constructible sets have Krull dimension?

Let $(I,\leq)$ be a poset. Recall that the Krull dimension of $I$ is defined as follows: -- $K.dim(I)=-1$ if and only if $I=\{0\}$; -- if $\alpha$ is an ordinal and we already defined what it means ...
3
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1answer
114 views

Classification of maximal nonuniform Fuchsian lattices existent?

I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the ...
3
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1answer
119 views

Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?

Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...
4
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1answer
289 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
4
votes
1answer
140 views

Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices

I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to ...