Questions tagged [dimension-theory]
Hausdorff dimension, box dimension, packing dimension and similar concepts.
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The Lebesgue covering dimension of the Cosmic String interval topology
Take the spacetime $(M,g)$ that satistfies Einstein's Field Equations exactly where $g$ is locally:
$$g=-(cdt-a d\phi)^2 + d \rho^2 + \kappa^2 \rho^2 d \phi^2 +dz^2 \ , \ \ \kappa>0 , \ a\in \...
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N dimensional, not-locally Euclidean, non-Hausdorff topological space
Take a topological space $(M, \tau) $ where $\tau$ is the collection of open sets of $M$.
Suppose:
the Lebesgue covering dimension of this space is $N \geq 1$
Non-Hausdorff
Not locally Euclidean
The ...
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What is the Lebesgue covering dimension of this topological space?
Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal.
Take the induced topology defined by the Lorentzian metric called Alexandrov topology.
This topology matches ...
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Box dimension and graph of Hölder function
In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that,
if a function $f:I^{d}\...
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Existence of finite 3-dimensional hyperbolic balanced geometry
Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
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Conformal welding and Jordan loop consequences?
In the similar context as Conformal welding of rectifiable curves
In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ ...
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Local dimension of stationary measures for iterated function systems with an expanding map
Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P = (p/2,p/2,1-p),$ where:
$f_1,f_2: I\to I$, where $...
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How far can one get by counting spaces of solutions this way?
I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the ...
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Dimension of a subspace of $n\times n$ real symmetric matrices
Let $n\in \mathbb N.$ Let $W$ be a non-trivial subspace of $n\times n$ symmetric matrices such that for every $x\in \mathbb R^n\setminus \{0\}$ there exists $a_x\in \mathbb R^n\setminus \{0\}$ such ...
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$\sigma$-product of the Hilbert cube
Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$
("eventually&...
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Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
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Does the pseudo-arc contain Erdős space?
The pseudo-arc is the unique hereditarily indecomposable chainable continuum.
The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense ...
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Can you remove a zero dimensional subspace from a cube and obtain a planar space?
The question, which came up in a conversation with my advisor Ola Kwiatkowska, is pretty much in the title:
Let $Z\subseteq[0,1]^3$ be zero-dimensional. Is it possible for $[0,1]^3\setminus Z$ to be ...
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Iterating the dimensional kernel of a metric space
Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let
\begin{align}
\Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\
\Lambda^2(X)&=\Lambda(\...
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The real dimension of any real algebraic set equals the complex dimension of its complexification
I want to prove the following statement. Please help!
Given any semialgebraic set $A$, consider its real Zariski closure $V_{\mathbb{R}}$ (which always has the same real dimension of $A$). Now ...
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A generalized Hausdorff dimension in form of a Lower semi continuous function
Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
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Seeking for references - Bowen Formula and a link between dimension theory and thermodynamic formalism
I'm needing references - preferably published papers and books - about this subject. I'm relatively new to the state of the art of fractal geometry and am way too inexperienced to seek for myself at ...
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Why do almost all points in the unit interval have Kolmogorov complexity 1?
Re-posted from math.stackexchange as I did not get any answers there.
I am reading
Jin-yi Cai, Juris Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line, ...
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Dimension of sumset vs sum of dimensions
Let $A\subset \mathbb R$. Is it true that
$$
\dim(A+A)\le 2\dim A
$$
for some dimensions – say, lower box for the LHS and upper box for the RHS.
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Commutative Banach algebras with zero-dimensional maximal ideal space and disjoint supports of Gelfand transforms
Let $A$ be a commutative semi-simple unital Banach algebra and let $\Delta$ be the maximal ideal space of $A$. Denote by $\widehat{\cdot}\colon A\to C(\Delta)$ the Gelfand transform.
If $\Delta$ is ...
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Does finite Hausdorff dimension imply finite packing dimension?
In other words, does there exist a metric space $(E,\rho)$ with finite Hausdorff dimension but infinite packing dimension?
Here are my thoughts:
I know that it is generally hard to relate Hausdorff ...
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Dimension properties of some concrete hereditarily disconnected subspaces of the Hilbert cube
This question was motivated by this MO-question asking about the example of a hereditarily disconnected metrizable separable space, which is not the union of countably many totally disconnected ...
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Is there a hereditarily disconnected space which is not the union of countably many totally disconnected subspaces?
A topological space $X$ is called
$\bullet$ totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$;
$\bullet$ ...
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Classification of Polish spaces up to a $\sigma$-homeomorphism
A function $f:X\to Y$ between topological spaces is called
$\bullet$ $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
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Is there a universal totally disconnected Polish space?
A topological space $X$ is called totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$.
In 1973 Roman Pol proved that ...
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Dimension-preserving non-linear map
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\...
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Sufficient conditions for the covering dimension and large inductive dimension of compact Hausdorff spaces to coincide
I have been looking through Alan Pears' "Dimension theory of general spaces" recently. In this book Pears references a 1960 paper by Aleksandrov and Ponomarev called "Some classes of $n$...
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Fraction dimensional "Euclidean" space
The “dimension” of Euclidean space $\mathbb{E}^n$ can be explained as an algebraic property, simply as a dimension of a vector space over the field $\mathbb{R}$. It also can be understood as a ...
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Sequences with 0's in $\mathbb R ^\omega$
Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology.
Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0.
Let $Y$ be the set of ...
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Dimension of Cartesian products
Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have
$$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this ...
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Embedding CW-complexes into infinite-dimensional topological vector spaces
Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
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Increasing a nowhere dense set in $\mathfrak E_{\mathrm{c}}$
Let $X$ be a closed nowhere dense subset of the complete Erdos space $$\mathfrak E_{\mathrm{c}}=\{x\in \ell^2:x_n\notin \mathbb Q\text{ for all }n<\omega\}.$$
Can you always find a closed nowhere ...
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Equidimensional Morphism
I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition:
Definition 2.1.2.
A morphism of schemes $p:X\rightarrow S$ is ...
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Isotopy Classes and Embeddability of Products in $\mathbb{R}^2$
On MSE I asked if the plane contains an uncountable collection of mutually disjoint copies of the Warsaw Circle; it seems to be false, and is probably already known but I'm not sure that anybody has ...
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Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
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Existence of stable spaces
An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space ...
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Subset of $\mathbb R$ with equal Fourier, Hausdorff and Minkowski dimensions
It is a standard fact that for $0\leq s\le1$, there is a compact set $C\subseteq [0,1]$ with Hausdorff and Minkowski dimensions $s$ (by modifying the construction of a Cantor set).
It is also a ...
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Can someone please help me understand the concept of twins? [closed]
I am unable to understand Lemma 2.3 of
Carmen Hernando, Mercè Mora, Ignacio M. Pelayo, Carlos Seara, David R. Wood, Extremal Graph Theory for Metric Dimension and Diameter, Electronic J. ...
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Which metrizable spaces can be embedded into the countable power of $\omega$ with the cofinite topology?
Let $\omega_{cf}$ be the countable space $\omega=\{0,1,2,3,\dots\}$ endowed with the cofinite topology $$\tau_{cf}=\{\emptyset\}\cup\{U\subseteq\omega:\omega\setminus U\mbox{ is finite}\}.$$
It is ...
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Minkowski (box-counting) dimension of generalized Cantor set
I'm trying to solve this problem.
For $0<\alpha, \beta<1,$ let $K_{\alpha, \beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{\alpha, \beta}^{0}=[0,...
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Is the Hilbert cube the countable union of punctiform spaces?
Recall that a (separable) metric space is called punctiform, if all its compact subspaces are zero-dimensional. While "natural" spaces would seem to be punctiform if they already themselves ...
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How to understand the "boundary" of subscheme, as defined in "An elementary characterisation of Krull dimension"
In An elementary characterisation of Krull dimension and A short proof for the Krull dimension of a polynomial ring, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, ...
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A natural $\mathbb Q\times \mathbb P$ subset of $\mathbb R$?
I would like a simple description of a dense subset of $\mathbb R$ which is homeomorphic to $\mathbb Q\times \mathbb P$. Preferably the description will be of an algebraic nature, and perhaps the set ...
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What is the analog of the symmetrized Jacobi matrix for delay equations?
For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...
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$\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$
I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
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Unknown work of Nöbeling on topological/Hausdorff dimension
Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$.
A well known result of
Szpilrajn (He changed his name to ...
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What is special to dimension 8?
Dimension $8$ seems special, as the partial list below might indicate.
Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$?
Surely it isn't it, in the end, simply because ...
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Perfect images of complete Erdős space
Let $\mathbb P$ denote the space of irrational numbers. In an answer to this question, Taras Banakh showed that the perfect images of $\mathbb P$ are precisely the Polish spaces with no compact ...
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Quotients of the irrationals
Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
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Fourier dimension of radial set
In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...