Questions tagged [vector-lattices]

For questions about vector lattices (aka. Riesz spaces; a type of vector space equipped with a partial order), Banach lattices and similar topics.

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How to demonstrate the finite height of a lattice?

I would like to ask you for help with a formal demonstration concerning the finite height of a lattice. My lattice is defined like this: is a lattice of vectors, each with exactly $n$ cells. In each ...
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94 views

Closed Paths on Hexagonal Lattice

I am trying to prove the following asymptotic formula for $a_n$, the number of closed paths of length $n$ on the hexagonal lattice (paths starting and ending at the same hexagon): $$a_n\sim \frac{6^n \...
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Calculating the fundamental parallelepided of a lattice space

Here, definition 4, the funademental parallelepided can be seen to be calculated by: $\sum_{i=1}^{i=n}x_i\mathbf{b_i}:0\leq x<1$ Where $\mathbf{b_i}$ is the basis of the lattice space. However, ...
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What is the number of shortest vectors of the nine dimensional lattice with basis vectors…

What is the number of minimal vectors for the nine-dimensional lattice (in $\mathbb{R}^{64}$) formed by taking integer combinations of the row vectors below? Edit: See pastebin for matrix. The ...
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Proof of additivity of the positive linear functional $\phi^+$ on a vector/Banach lattice that will be $\phi\vee0$.

For context, this is used in defining $\phi\vee0$ in a proof that the dual of a vector lattice is a vector lattice. Given a linear functional $\phi$ on a vector lattice $V$, define $\phi^+$ on $V^+$ ...
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Math theory that deals with ordered attr-value items?

There is partially ordered sets and lattices. Is there a branch of math that deals with ORDERED Attribute-Value items/objects. F.e. av-items /see that attrs also can be missing i.e. doors&roof/ : ...
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32 views

The number of lattice points inside R-radius ball centered at origin

This is a question from Hoffstein cryptography book. I'm trying to show that $\lim_{R \to \infty}$$\frac{{\#(\mathbb{B}_R(\mathbf{0})}\cap L)}{{Vol(\mathbb{B}_R(\mathbf{0}))}}$$=\frac{1}{Vol(\...
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closed lattice ideal is isomorphic to $C(K)$

Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
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Is there a lattice approximation of a double-covered space?

We can approximate a manifold with points on a lattice. For a space which has the symmetries of a double-covered space. (i.e. you have to rotate 720 degrees before you end up at the same place.), is ...
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Regular tilings of n-simplex

Consider a regular n-simplex (the n-dimensional generalisation of a triangle/tetrahedron). A triangle will tile the plane in a triangular pattern. In 4, 8 and 24 dimensions. Can we tile the volume ...
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Cocompact & discrete lattice

I don't understand a step in the proof of the following proposition: Let $\Lambda$ be a subgroup of a real vector space $V$ of finite dimension. Then $\Lambda$ is a full lattice if and only if $\...
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Establish Archimedean property of a vector-lattice

I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem. I feel the statement below (or in fact weaker versions) should ...
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About the infimum of two operators in vector lattices

A real vector space $E$ is said to be an ordered vector space whenever it is equipped with an order relation $\ge$ that is compatible with the algebraic structure of $E$. A Riesz space is an ordered ...
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About modulus of a vector in a vector lattice

A real vector space $E$ is said to be an ordered vector space whenever it is equipped with an order relation $\ge$ that is compatible with the algebraic structure of $E$. A Riesz space or vector ...
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A question about Riesz spaces

A real vector space $E$ is said to be an ordered vector space whenever it is equipped with an order relation $\ge$ that is compatible with the algebraic structure of $E$. A Riesz space is an ordered ...
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Almost sure and order convergence are equivalent in $L_p$ spaces

I have attempted to prove the following lemma, which is given as an exercise by Aliprantis and Border. Is the proof correct? Improvements are welcome. Lemma. An order bounded sequence $(f_n)$ in some ...
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48 views

second theorem of Minkowski proof

I am wondering if anyone have a proof of second theorem of Minkowski. He says that if $vol(K) = 2^{n} det(L)$ and $K$ is compact and symmetric and convex then $K$ contains a non zero lattice point. I ...
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example of a partially ordered vector space

Are there any well known examples of partially ordered vector spaces (Riesz Spaces) from applied or pure mathematics?
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35 views

Does Dedekind completeness imply the Archimeadean property?

Suppose that $A$ is a Riesz space (vector lattice). Recall the following terminology: A is Archimedean if for any $x,y\geq 0$ such that $n x\leq y$ for $n=1,2,\ldots$ is follows that $x=0$; A is ...
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105 views

When does projecting a lattice onto a subspace give another lattice?

For a lattice $L$ and vector $v$ in $\mathbb{R}^n$ take $L\backslash v := \{l-(\textstyle \frac{v^\dagger l}{\|v\|^2})v: l\in L\}.$ When is $L\backslash v$ a lattice?
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About relatively uniform convergence in Riesz spaces

A Riesz space is a partially ordered real vector space such that every two elements has a sup and indeed it is a vector lattice. I have encountered to the definition of relatively uniform convergence ...
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definition of complete vector lattice

Suppose $M$ is a von-Neumann algebra, $L=M\cap M'$ is the centre of $M$. The last line on page 29, C*-algebras and their automorphism groups, states that the self-adjoint part $L_{sa}$ of $L$ is a ...
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A simple modulo arithmetic problem

I slur `$z\mod L$' here to mean the only element of $\{z+nL: n\in \mathbb{Z}\}\cap [0,L).$ We are given quantities: $a,b, L,$ $D_1 = (ax \mod L) + aw$, $D_2 = bx+bw.$ We are also given the fact: ...
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Expected number of lattice points in $S$ is like $S$'s volume

Hel${}$lo everyone, I think I remember a theorem going something like this: $L$ is a lattice in $\mathbb{R}^n$ whose base cell has area $a$. $S$ is a bounded open set in $\mathbb{R}^n.$ Take an ...
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A question about the supremum and infimum in a Banach lattice

Given a Banach lattice $X$, it is well know that every finite subset $A$ of $X$ has a supremum and an infimum. For example, if $\{ x_1, \ldots, x_n\}$ is a finite subset of $X$, then we have $$ sup (...
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A function space satisfying $ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta| $

Consider a class of function on the real line such that $$ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta | $$ for a $\theta >0$. Does this class of function space have a name When $ |f(x) -f(y)| \le ...
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Functional inequality on $\mathbb{Z}^d$

Let $B_L = \{ x \in \mathbb{Z}^d : |x| < L\}$ be a box of side length $L \in \mathbb{N}$, where $| \cdot|$ is the $L_{\infty}$ norm and and assume $d \geq 3$. Let for any $L \in \mathbb{N}$ , $f_L :...
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60 views

Why does dot product give correct angle, but cross product does not?

Question: The primitive lattice vectors for a body-centered cubic lattice are given by $\vec a_1 =\frac{a}{2}(1,1,-1)$ , $\vec a_2 =\frac{a}{2}(-1,1,1)$ , $ \vec a_3 =\frac{a}{2}(1,-1,1)$. Calculate ...
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Lattices in $\ell^\infty(X)$

Let $X$ be a set and let $A$ be a subspace of $\ell^\infty(X)$. Prove that for any $f,g \in \ell^\infty(X)$, $f+g = f \vee g + f \wedge g$ and $|f-g| = f \vee g - f \wedge g$. Show that $A$ is a ...
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Vector lattice and sublattice

Consider the integer lattice $(\mathbb{Z}^2,\leq)$ with the canonical partially ordered relation $(a,b)\leq(c,d)$ iff $a\leqslant c$ and $b\leqslant d$. Now just consider a partially ordered subset of ...
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Riesz space of functions from any set $X$ to $\mathbb{R}$

We collect certain functions from $f:X\to\mathbb{R}$ to $\mathscr{T}$ so that it is a vector space over $\mathbb{R}$ under the usual function addition, scalar product. Now we define an partial order ...
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Why is the lattice decoding problem NP-hard?

Given an alphabet $\mathbb{A}$ and a lattice $L = \{Y : Y = Hx, X \in \mathbb{A}^n\}$, the lattice decoding problem is to find: $$ \hat{x} = \text{argmin } \| Z - Hx \| $$ Where Z is some vector $\...
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Is there a list of the couples of lattices commensurate with each other?

Consider two periodic lattices of points extending to infinity and the question of commensurability of the two. If for instance we take one of them to be a square lattice called $S_1$, it is fairly ...
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Lattice of vectorial space

Let $V$ be a vectorial space and $L(V)$ a lattice of the subspaces of $V$. Show that $L(V)$ is distributive if and only if $dim(V)=1$. I don't even know how to start this problem. Any hint would be ...
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Lattice diagram

First of all I'm new to Lattice and hasse diagrams theory. I don't understand why this diagram is not a Lattice. Supposedly this is not a Lattice because not all elements have a join. Why can't $f$ ...
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273 views

Quotient of lattices and volumes

Let $V$ be a finite dimensional $\mathbb R$-vector space. Suppose that $L\subset V$ is a lattice (i.e. a discrete subgroup) and $M\subset L$ is another lattice, which is also a subgroup of $L$. ...
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“Explaining” a coincidental connection between Pascal's triangle and symmetries of lattices

The $1$s on the edges of Pascal's triangle appear infinitely many times in the array. One number ‒ just one ‒ appears exactly once. Infinitely many have multiplicity $2.$ Infinitely many have ...
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Show that two projections commute if and only if their respectively closed subspaces are compatibles

I have the next problem: Let $\mathcal{H}$ be a separable and complex Hilbert space, with $S$ and $Q$ two closed subspaces on it. Let $P_S$ and $P_Q$ be the orthogonal projection onto $S$ and $Q$ ...
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Is a linear operator that is tangent to a continuous function from a TVS to a Riesz space also continuous?

Let $\mathcal{X}$ be a topological vector space and $\mathcal{Y}$ be a topological Riesz space. Suppose $f:\mathcal{X} \rightarrow \mathcal{Y}$ is continuous at $0_{\mathcal{X}}$ and has $f(0_{\...
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Whom is the decomposition theorem for finite-dimensional Riesz spaces (vector lattices) is attributed to?

There is a result in the theory of Riesz spaces suggesting that every such finite-dimensional space over R, is isomorphic to a direct sum of orthogonal ideals, each one being a lexicographically ...
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106 views

Are the two given lattices related to each other by a simple rotation?

I am trying to come up with a general algorithm to determine if two lattices are related to each other by a simple rotation operation. The way I think about a lattice is as an array of points. The ...
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“Representation Capacity” of Finite Lattice Ordered Modules

Apologies for what is probably a very basic question, but I am looking for references for what sort of lattices can be represented in what I think should be called something like "finite lattice-...
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Example of a case that shows that a Hilbert lattice of infinite dimension is not modular.

Let $H$ be a separable Hilbert space. Let $C(H)$ be the set of closed subspaces of the Hilbert space $H$. Let $\le$ be the inclusion $\subseteq$. For $a,b \in C(H)$ define $a^\perp$ as the ...
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Quotient Riesz space given by a filter $\mathscr{F}$ is Archimedean iff $\mathscr{F}$ is closed under countable intersections.

The following is given as Example 1XE in Fremlin's Topological Riesz Spaces and Measure Theory. Let $X$ be a nonempty set, $E = \mathbb{R}^X$, and $\mathscr{F}$ be a filter on $X$. Let $$F = \{x \in ...
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Dual space of $n$-simplex tiling

Let us consider the dual space of $n$-simplex tiling. Here the dual space is defined by replacing the $d$-dimensional object $p$ by the co dimension $n-d$ objects at the symmetric center of $p$. It is ...
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$\mathbb{R}^d / \Gamma$ where $\Gamma$ is an irrational lattice

A lattice $\Gamma \subset \mathbb{R}^d$ is said to be rational if for any two vectors $\gamma_1, \gamma_2 \in \Gamma$, their inner product satisfies the relation $$\displaystyle \langle\gamma_1, \...
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Bounded continuous Riesz isomorphisms

I want to know if $C(X)$ and $C(Y)$ are Riesz isomorphic is it true that $BC(X)$ (the space of bounded continuous functions on $X$) and $BC(Y)$ also have to be Riesz isomorphic. I already found the ...
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How to find and odd number $n$ s.t. $nV\subset W$?

Theorem 3.6 i) here (Pratulananda Das and Ekrem Savas: On $I$-convergence of nets in locally solid Riesz spaces, Filomat 27:1 (2013), 89–94, DOI:10.2298/FIL1301089D) $3.6$ i) $I_τ-\lim s_α = x_0 ⇒ ...
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Is the sum of i.i.d. discrete Gaussian rv still distributed as a discrete Gaussian?

Let assume I have $n$ i.i.d. random variables $X_i \; i=1,\dots,n$. Each of them is distributed as $X_i \sim \mathcal{D}_{L,\sigma,c}$, namely a discrete Gaussian distribution over a Lattice $L$ of ...
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Minimal turns to cover points of a 3D lattice

It is possible to cover the points of a size 4 cubic lattice with a closed loop of 28 turns, as seen in the images below. In these solutions, the turns only occur at the given points, and no point ...