Questions tagged [dimension-theory]
Hausdorff dimension, box dimension, packing dimension and similar concepts.
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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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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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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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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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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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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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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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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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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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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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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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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 this ...
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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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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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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 Hausdorff)...
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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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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, i....
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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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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 ...
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Hausdorff dimension of subsets of the Mandelbot set.
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper, but I can't figure out one thing : can we say all open subsets of this boundary has ...
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Simple definition of the Hausdorff measure using squared paper
I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...
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Analogues of Luzin's theorem
If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of $L^\infty(X,\mu)$,...
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Rank of a module
What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...
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Gaps in Dimension Polynomials
There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a ...
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Dimension of tensor product of modules
$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ {B}(M\...
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Why do modules with small support have high Exts?
Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The ...
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Converse of Principal Ideal Theorem
$(A, \mathfrak{m})$ a Noetherian local ring, $a\in\mathfrak{m}$ a zero divisor. Then is it true that $\mbox{dim}\ A/(a) = \mbox{dim}\ A$ ?
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Dimension of module
Does dimension of a module (say, dimension of its support) have anything to do with the supremum length of chains of prime submodules like rings?
Let's restrict to finitely generated modules over ...
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How big can the Hausdorff dimension of a function graph get?
This question is inspired by How kinky can a Jordan curve get?
What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the ...
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Dimension of subalgebras of a matrix algebra
If $n$ is given and $A$ is a subalgebra of $M_n(\mathbb C)$, the algebra of $n \times n$ matrices with entries in the field of complex numbers, then what are the possible values of dimension of $A$ as ...
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how slow can the dimension of a product set grow?
Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes "...
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Determining a lower bound on the Hausdorff dimension of a set
Does anyone know of a good method for finding a lower bound of the Hausdorff dimension of a set $G$?
The only method I could find is to find an $\alpha$-Hölder function $f \colon G \to H$ then $\...
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Different definitions of the dimension of an algebra
I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...
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