Questions tagged [dimension-theory]
Hausdorff dimension, box dimension, packing dimension and similar concepts.
183
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Topological spaces with Lebesgue covering dimension 1
We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general
connected topological space with the ...
2
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0
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What is a precise definition of a twisted fibration of one fern over another?
I am curious if there is a notion of a "twisted fibration" of fractals. Since there are many classes of fractals, I'll ask specifically about L-systems.
How can we precisely define the twisted ...
9
votes
1
answer
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Box dimension of the set of Pisot numbers?
A Pisot number is an algebraic integer bigger than $1$ with all of its Galois conjugates having modulus less than $1$. The set of Pisot numbers is known to be countably infinite and is not dense in $(...
6
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5
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Fractals of dimension zero
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?
I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
5
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1
answer
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Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
6
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0
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What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
7
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2
answers
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Haar measure on the Grassmannian space
The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $...
6
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1
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Random Cantor sets on the unit interval
Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$.
For ...
5
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0
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225
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Do $G_\delta$-measurable maps preserve dimension?
This question (in a bit different form) I leaned from Olena Karlova.
Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...
8
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1
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Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?
Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the ...
11
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1
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Dimension in CW-approximation
The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question.
Let $X$ be a topological space, and let $\tilde{X}\to X$ be a CW-...
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Two definitions of Lebesgue covering dimension
Maybe this question has already been considered here, but after a quick search I didn't find what I was looking for.
As I see, in the literature there are two different definitions of the ...
6
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1
answer
329
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Factorization of a certain map through a CW-complex
Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...
9
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1
answer
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$U_q(\mathfrak{sl}_2)$ representations of "quantum dimension" zero
I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I ...
3
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0
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What is the connection between the Riemann Xi-function and n-sphere? [closed]
Riemann's Xi-function is defined as
$$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$
At the same time we have the following formulas for n-sphere's area and volume:
$$\begin{array}{...
3
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1
answer
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Sort shapes in 4 dimensions [closed]
I'm sure you know the baby game about sorting shapes by putting them in holes:
I'm wondering if such a game could exist in 4 dimensions? I imagine the shapes would have 4 dimensions and the holes 3 ...
4
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1
answer
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Bound for the generalised Rényi dimension of a measure
If $\mu$ is a measure on $\mathbb{R}^d$, and for each $r>0$ we let $\mathcal{M}_r$ denote the set of all ``cubes'' of the form $[j_1r,(j_1+1)r) \times \cdots \times [j_dr,(j_d+1)r)$ for $j_1,\ldots,...
11
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Self-Similar Graphs
Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs.
Many definitions of fractal dimensions (...
7
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3
answers
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How can dimension depend on the point?
Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
4
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1
answer
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Existence of small projective dimensioned modules
Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$.
Then do either of the ...
11
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1
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Geometric measures different from Hausdorff
$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$
In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset \RR^n$...
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0
answers
245
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Global dimension of a subalgebra with all units
(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit in $...
13
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3
answers
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Dimensions of self-affine sets
Let $A$ be a $2\times 2$ matrix which we assume to be contracting, i.e., the exists $\alpha\in(0,1)$ such that
$$
\|A {\mathbf x}\|_2\le \alpha\|{\mathbf x}\|_2,\quad \forall {\mathbf x}\in\mathbb R^...
3
votes
1
answer
661
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Hochschild cohomology and formal smoothness
Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological ...
11
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1
answer
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Why do convex polytope options constrict with dimension, rather than expand?
There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There ...
5
votes
1
answer
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Multifractal Analysis and Dimension Spectrum of Unions
Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets
$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{...
5
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2
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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 ...
5
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1
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Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?
I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
4
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Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets
Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
7
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2
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Arithmetic products of Cantor sets.
Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product
$AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are self-...
2
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2
answers
166
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A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line
I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a ...
2
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0
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Is there a better function (linear or even a projection)?
Let $A$ be a finite $n$-element set. Let $\mathbb R^A$ be an $n$-dimensional Euclidean space (with the ordinary Euclidean distance). Let $X$ be an arbitrary topological space. Consider a continuous ...
1
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0
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When does the rank of a module behave sub-multiplicatively under tensoring?
Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product
$
\cal{E} \otimes_A \...
4
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2
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Hutchinson's formula for asymptotically homogeneous Cantor sets
As everyone knows, the standard middle-thirds Cantor set is constructed by dividing the interval into three equal parts, removing the middle one, then applying the same procedure to the remaining two ...
3
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2
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Can the isoperimetric dimension of a d-generated group attain any value?
Background
The isoperimetric dimension of a finitely generated group $G$, which we denote by $\dim(G)$, is the largest number $d$ such that any Cayley graph $\Gamma$ of $G$ (constructed with respect ...
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3
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Zero-dimensional space
Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that $\overline{A}=\overline{\...
7
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2
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On some finiteness properties for schemes
Consider the following properties of scheme $X$:
A: $X$ is of finite type over $\mathbb{Z}$
B: $X$ is Noetherian
C: $X$ is of finite Krull dimension
What implications are there between these three?...
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6
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Pathological Examples of Dimension
I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
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5
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Tessellating $\mathbb{R}^n$ by bricks.
Let us call the $\ell_1$-product of intervals $[0,k_1]\times...\times [0,k_n]$ a brick of size $k_1+...+k_n$. Consider a tessellation $T$ of $\mathbb{R^n}$ by (shifted) bricks so that every point ...
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infinite dimensional CAT(0) groups
Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and ...
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1
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existence of fractal [closed]
I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$?
If yes, do we have any method to construct such ...
14
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5
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Is there an axiomatic approach of the notion of dimension ?
There are many notions of dimension : algebraic, topological, Hausdorff, Minkowski... (and others).
While the topological one generalize the algebraic one, the last three need not coincide for every ...
32
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3
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Krull dimension less or equal than transcendence degree?
Let $k$ be a field, and $A$ a $k$-domain, so that the fraction field of $A$ has transcendence degree $n$ over $k$.
If $A$ is finitely-generated over $k$, then $A$ has Krull dimension $n$ (Theorem A ...
6
votes
3
answers
663
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Minimum space dimension to place n-points knowing pairwise distances
Let $P$ be a set of $n$ points.
Assuming I know the pairwise distances for each pair of points.
What would be the minimum dimension of the space in which I could place those $n$ points with respect to ...
5
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3
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dimension of a real affine variety
Let $V$ be a real affine variety in $\mathbb R^n$, i.e. the zero set of a real polynomial $p(x_1,\dots,x_n)$. Consider the following three definitions of the dimension of $V$, $dim(V)$.
Definition ...
20
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2
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A "dimension" for Tychonoff spaces
It's well-known that any Tychonoff space $X$ can be embedded in $[0,1]^k$ for some cardinal $k$. It's natural to ask what the smallest such $k$ is (let's call it $k(X)$). However, this probably ...
4
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0
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234
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dimension of induced comodule
Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...
4
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Hausdorff dimension of graphs .
Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
3
votes
2
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357
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Hausdorff dimension of inverse images.
Let $f: \mathbb{R}^d \to \mathbb{R}$ be a continuous function. Let $t \in (\inf(f), \sup(f))$ and define $C = f^{-1} (t)$. Is it true that the Hausdorff dimension of C is $\geq d -1$? If no how does ...
9
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1
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When does the homological dimension of a tensor product equal the sum of dimensions?
The notion of dimension I prefer most is right global dimension, but the question can also be asked for other notions (e.g. weak dimension, injective dimension, Krull dimension). Letting $d$ be ...