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])
652 questions
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Shortest vector problem over polynomials
In shortest vector problem, given a lattice in $\Bbb Z^n$, we seek the shortest non-zero vector in the lattice. This problem is computationally difficult.
Answer in Evidence for integer factorization ...
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0
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231
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Writing integers in ring of integers of number fields
Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...
user76479
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1
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Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?
$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times.
I am studying some function arising from symplectic geometry which happens in my case to be ...
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3
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2-dimensional sublattices with all vectors having very big square (in absolute value)
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not
definite, not necessarily unimodular, $n>2$. I want ...
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2
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Intersection of two lattices
Suppose that $\Lambda_1, \Lambda_2$ are two sub-lattices of $\mathbb{Z}^n$ of full rank, defined by congruence modulo a prime $p$. That is, there exist two vectors with integer entries $\mathbf{a}, \...
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425
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Probability a random matrix contains a short integer vector in its kernel
Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
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2
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Permutation covering of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
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Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?
Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$)
Or is it known that it cannot be such a lattice ?
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Is the product of two supermodular functions supermodular?
The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have
\begin{equation*}
f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′).
\end{equation*}
Suppose $f$ and $g$ are supermodular, non-...
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answer
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Number of matrices with given Smith normal form
Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
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1
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Bounding the number of lattice points inside an $n$-dimensional ellipsoid
I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse.
That is, given an $n$-dimensional ellipsoid ...
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5
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2
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Ordered lattice point enumeration
I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community.
Setup: Let $e_{...
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2
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covering radius of a lattice from cyclotomic extension
Given a finite field extension $L$ of $\mathbb{Q}$ of dimension $n$, there is a natural way to embed it into $\mathbb{R}^n$ such that the image of its ring of integers $\mathcal{O}_L$ is a lattice. If ...
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When do two lattices have the same stabilizer in the diagonal torus?
This is moved from MSE, where I asked and didn't receive an answer (see https://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...
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Can any finite distributive weighted lattice be realized by inclusion of groups?
By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice.
A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...
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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 : U]...
- 11.3k
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answer
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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 = \{x\in\...
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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 ...
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1
answer
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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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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, ..., \mathbf{...
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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 ...
0
votes
1
answer
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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 $(1,T)$. ...
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0
answers
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$\mathbb{Q}$-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?
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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 ...
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votes
3
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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, ...
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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 ...
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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 ...
- 3,513
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0
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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?
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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 ...
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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 ...
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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 $d$-...
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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 ...
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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 ...
user39066
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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 = \...
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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 ...
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1
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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:
...
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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 $$a \oplus b = \frac{1}{\frac{1}{a}...
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Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$
$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying
$b$ is bilinear,
$b(x,...
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What's the ratio of inclusions of finite groups with a distributive lattice?
Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
...
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2
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Covolume of the row span of a matrix and of the kernel of a matrix
Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The covolume
$\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a
fundamental domain for $L$, i.e., the volume of the parallelopiped
...
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1
answer
1k
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Examples of fundamental domains
Everyone knows that it's difficult to compute a general fundamental domain for an arithmetic group but are there any specific examples where such domains have been calculated? I'm mostly interested in ...
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2
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562
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Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$.
It is well known that in dimension ...
- 2,521
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A question on the poset of classes of isomorphic subgroups of finite groups
Given a finite group $G$, we consider the set $${\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
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0
answers
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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 ...
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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 ...
- 156k
4
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1
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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 ...
- 320
2
votes
1
answer
375
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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 $(X,\le)$?
- 1,145
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1
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121
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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 \...
- 320
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0
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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 ...
1
vote
1
answer
259
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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 single ...