Hausdorff dimension, box dimension, packing dimension and similar concepts.

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86 views

### 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 ...

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209 views

### 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 ...

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**3**answers

265 views

### 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 ...

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**1**answer

210 views

### 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 ...

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**1**answer

208 views

### 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 ...

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83 views

### 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) ...

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446 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 ...

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129 views

### 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 ...

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82 views

### 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 ...

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**1**answer

281 views

### 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 ...

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139 views

### 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 ...

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143 views

### 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 ...

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102 views

### 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 ...

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161 views

### 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 ...

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**1**answer

117 views

### 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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763 views

### 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 ...

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**2**answers

396 views

### 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 ...

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**6**answers

1k views

### 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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**4**answers

528 views

### 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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**2**answers

607 views

### 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**answer

320 views

### 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 ...

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**5**answers

876 views

### 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 ...

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2k views

### Krull dimension <= 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 ...

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372 views

### minimum space dimension to place n-points knowing pairwise distances

Hi everyone,
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 ...

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603 views

### 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 ...

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**2**answers

486 views

### 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 ...

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**0**answers

176 views

### Decompose a set into sets of Hausdorff-dimension n-1

Assume we can decompose a set $A$ in $\mathbb{R^n}$ of Hausdorff-dimension n into sets $(A_t)$ $t\in [0,1]$ of Hausdorff-dimension n-1 whose n-1-dimensional volume is known (for example is zero).
...

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187 views

### dimension of induced comodule

Let $\pi : G --> H$ 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 ...

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609 views

### Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?

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236 views

### 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 ...

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**1**answer

514 views

### 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 ...

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154 views

### Is there a relationship between the right global dimensions of R and R[1/v]?

A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...

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3k views

### What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...

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763 views

### Can we say anything about the Krull dimension of a localization?

I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...

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**0**answers

168 views

### Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...

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**2**answers

163 views

### Dimensionality of a map — distance

Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further.
FORMAT
...

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404 views

### Quantitative measurement of infinite dimensionality

I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy ...

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**1**answer

146 views

### systems of parameters vs. minimal “exhausting” systems in a Noetherian local ring

Hello,
Probably this is a very easy question.
Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$.
Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / ...

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**0**answers

81 views

### Hausdorff Dimension of non-locally maximal hyperbolic sets

We're referencing Yakov Pesin's "Dimension Theory in Dynamical Systems" in an effort to compute the Hausdorff dimension of a particular invariant set $\Lambda$ of a hyperbolic toral automorphism. ...

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**2**answers

486 views

### Non-Hölder continuous devil's staircases

Let $f:[0,1]\to[0,1]$ be a devil's staircase in the usual sense. (That is, $f$ is continuous, non-decreasing, $f'=0$ on a set of full Lebesgue measure.) We also require the complement to the set where ...

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676 views

### Topological dimension, is it local?

Let $n\in\mathbb N$ and $X$ be a complete metric space.
Assume that there is $\epsilon>0$ such that
$$\dim B_\epsilon(x)\le n$$
for any $x\in X$.
Is it true that $\dim X\le n$?
Here ...

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**1**answer

234 views

### Does the Hausdorff dimension depend on the L^p-norm?

A simple question from someone new to the field:
In a metric space, the Hausdorff dimension of a subset is defined by covering the subset with $\epsilon$-balls and looking at how the number of ...

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**3**answers

3k views

### Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to ...

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**1**answer

687 views

### Example in dimension theory

Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?

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2k views

### More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.
Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...

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**2**answers

290 views

### How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces?

Question 1. Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example ...

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**1**answer

389 views

### Can dividing out a group action can increase the Lebesgue dimension ?

Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$?
I know examples, ...

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**1**answer

252 views

### Lebesgue dimension of closures satisfying the Z-set condition

Given any subspace $A\subset X$ of a topological space with Lebesgue dimension $\le N$.
Let $\bar{A}$ denote the closure of $A$. Assume, that the pair $(\bar{A},A)$ satisfies the Z-set condition, ...

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**1**answer

189 views

### Hausdorff dimension of higher powers of the Mandebrot set ?

My third question about Shishikura's result :
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper1. The Mandelbrot set is defined by ...

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**2**answers

1k views

### Area of the boundary of the Mandelbrot set ?

My second question about Shishikura's result :
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it ...