Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [dimension-theory]

In general topology, dimension theory studies various notions of dimension defined for topological spaces, for example Lebesgue covering dimension, small and large inductive dimension or Hausdorff dimension. Notice that this tag is not intended for questions about dimension in linear algebra.

3
votes
0answers
85 views

Must a subset of $\mathbb{R} ^n$, that has a dimension of $1$, be formed by a union of lines?

Let $S$ be a subset of $\mathbb{R} ^n$ with a box-dimension of $1$. Is $S$ necessarily a union of lines (including "curved" lines and line segments)? If no, then is $S$ at least an infinite set of ...
3
votes
1answer
29 views

Covering dimension of boundary of compact subset of $\mathbb{R}^n$

Let $X$ be a compact subset of $\mathbb{R}^n$, with the inherited Euclidean topology. Does it follow that $\dim_{cov}(\partial X)\leq n-1$? I would be happy to have a reference for that, if it is ...
0
votes
0answers
41 views

The methods to create fractal objects mathematically [closed]

I have read that fractals does not have a mathematical definition as for now. So how do we understand a given object can be considered as a fractal? I know iterated function systems theory is a method ...
0
votes
0answers
34 views

About homogeneous dimensions of a Lie group

I am looking for a clear and simplified definition of the homogeneous dimension of a vector space or a Lie group. If anyone has the patience to explain me clearly his definition and if possible an ...
0
votes
1answer
30 views

Small Inductive Dimension is a topological invariant

I have started to read Engelking's book "Theory of Dimesions: Finite and Infinite" and the author states (pp. $3$) that: The small inductive dimension (also called Menger-Urysohn dimension) is a ...
0
votes
0answers
20 views

Decomposition theorem for topological dimension

I am trying to prove the decomposition theorem for the topological dimension: It is a known result (e.g. Engelking) that for a non-empty separable metric space $X$, the small inductive dimension $...
0
votes
0answers
38 views

Why are Hausdorff dimension and Minkowski (box) dimension not equivalent?

I am putting the finishing touches on my master's essay for graduation this semester and I want to end my paper with a proof of why Hausdorff dimension and Minkowski (box) dimension are different. ...
1
vote
1answer
33 views

Size of $\epsilon$-net on a $2$-dimensional compact manifold

I'm dealing with a $2$ dimensional compact Riemannian manifold on which I consider the distance induced by the metric. I would like to know the behavior of the size of an $\epsilon$-net when $\epsilon ...
0
votes
0answers
17 views

Does the Johnson–Lindenstrauss lemma require the Normal distribution?

The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly ...
1
vote
0answers
42 views

The box dimension of the graph of a continuous function is $\ge 1$

In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions: ...
1
vote
0answers
31 views

Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
0
votes
1answer
38 views

Is it possible to have a Hausdorff dimension less than the topological dimension?

"Normal" geometric shapes have Hausdorff dimensions equal to their topological dimensions. Mandelbrot defined fractals as shapes that have a Hausdorff dimension greater than their topological ...
0
votes
1answer
36 views

Hausdorff dimension of an open set

It seems that the following statement is pretty simple to prove: Let $\emptyset \neq M$ be an arbitrary open set in $\mathbb{R}^n$, where n is the smallest integer such that $M \subseteq \mathbb{R}^...
0
votes
0answers
33 views

What “dimensions” means in “Superstring theory has 10 dimensions”.

I read about Clifford Algebras and different Geometries and its relation to Lie Algebras and to String Theory. You have heard that a version of String Theory has 10 dimensions. Stuff like this make no ...
1
vote
1answer
37 views

Definition of codimension of variety

Let $X$ be an variety over field $k$. A Weil divisor on $X$ is an integral linear combination of irreducible subvarieties of $X$ of codimension $1$. So I want to know the definition of codimension of ...
1
vote
1answer
19 views

How to make a 5-cell-shaped maze?

This is a follow-up question to my previous question, which I felt was too unrelated to what I wanted: Expansion from 3D tetrahedron to 4D 5-cell via combining smaller parts Anyway, if you don't know ...
1
vote
1answer
44 views

Packing Dimension as a Countable Union of Minkowski Dimension Sets

Is it true that if $X$ has packing dimension $\alpha$, then we can write $X$ as the countable union of sets $X_i$, where $X_i$ has Minkowski dimension $\alpha$. If not, which notion of dimension is ...
1
vote
0answers
57 views

Hartshorne II.3.22c following the hint

I know this exercise can be solved in several ways, such as Ravi Vakil's proof or this answer, but I'd like to try to follow the given hint if possible. The exercise says the following: Let $f:X→...
2
votes
0answers
35 views

Dimension image of morphism of projective varieties

Let $f: \mathbb{P}^n \to \mathbb{P}^m$ be a rational map. Then there exists $U \subset \mathbb{P}^n$ open so that $f_{|U}$ is a morphism. What can we say about the dimension of $\overline{f(U)}$? We ...
0
votes
1answer
157 views

Hartshorne II-3.22(b)

Let $f:X\to Y$ be a dominant morphism of integral schemes of finite type over a field $k$. Let $e=\dim(X)-\dim(Y)$. For any point $y\in f(X)$, show that every irreducible component of the fibre $...
3
votes
0answers
27 views

Dimension of affine affine algebras as a module

Suppose that $A\cong \mathbb{R}[f_1,\dots,f_d]$ is a (commutative) affine $\mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $\mathbb{R}[x_1,\dots,x_N]$. When is $A$ a finite-...
2
votes
4answers
93 views

Meaning of Dimension

Is a (non-straight) curve in the xy-plane considered to be a 1-dimensional or 2-dimensional entity? Depending on the answer, what is the term used to describe its other attribute (either its oneness, ...
0
votes
1answer
31 views

Are 2D grayscale images actually 3D?

I keep running into this issue when describing the dimensionality of data I work with. In general it's a question of whether or not the 3D timeseries we are handling are in fact 5D, but I guess the ...
2
votes
1answer
38 views

a.c.c. and d.c.c. on radical ideals in commutative ring of dimension zero

Let $R$ be a commutative ring with unity of dimension zero (i.e. every prime ideal is maximal). Does any of the following two conditions imply the other : 1) $R$ satisfies a.c.c. on radical ideals ...
7
votes
0answers
84 views

Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
1
vote
1answer
32 views

Commutative Noetherian ring with distinct ideals having distinct index

Let $R$ be a Noetherian commutative, infinite ring with unity such that distinct ideals have distinct index i.e. if $I,J$ are ideals of $R$ and $I \ne J$ , then $R/I$ and $R/J$ are not bijective as ...
1
vote
0answers
38 views

Mandelbrot Boundary Area

I might be entirely off an my assumptions, but the following has led me to a question. The Mandelbrot set is contained by an border of infinite length. Said border is 2-dimensional. The Hilbert space-...
0
votes
1answer
47 views

Do strongly equivalent metrics have the same covering numbers?

Let $(X,d)$ be a metric space. Then for any $r>0$, the $r$-covering number of $X$ is the minimum number of open balls of radius $r$ needed to cover $X$. And if $d_1$ and $d_2$ be two metrics on ...
0
votes
1answer
30 views

Interpretation of $R^D$ (Vectors Space, Dimensions and Functions)

I'm aware that it's a trivial question, but I want to make sure that I'm understanding correctly what I'm studying, therefore I would like to ask you, can you tell me what does $R^D$ "say" in these ...
0
votes
0answers
46 views

Using 6th dimensional vector to rotate a tesseract

I'm trying to rotate a tesseract in 4D space for a project. This shows I can use bivectors to "to generate rotations in four dimensions." Following exterior algebra and finding the wedge product, I'...
2
votes
2answers
72 views

The Dimension of the Cantor Set: Clarification

I am reading Abbott's Understanding Analysis. He uses the following reasoning to conclude that the Cantor set has dimension 0.631. He asks the reader to consider the following informal definition of ...
4
votes
1answer
75 views

If $X \times X$ is normal, then is $X \times X \times X$ normal?

I am looking at some topological dimension theory for product spaces, and in trying to construct a certain type of counterexample it's become relevant to consider the question in the title above. I ...
1
vote
0answers
29 views

Axioms that characterize the notion of dimension [duplicate]

Let $\mathcal{C}$ be a class of sets/spaces/structures among which we have a dimension. Namely a map $d:\mathcal{C}\rightarrow \mathbb{N}$ defined in a certain manner that motivated the appellative ...
2
votes
0answers
27 views

Do compacta in $\mathbb R^n$ have finite Hausdorff measure? [duplicate]

For $A\subset \mathbb R^n$ let $\mathcal H^s(A)$ be the $s$-dimensional Hausdorff measure with respect to the Euclidean metric. The Hausdorff dimension of $A$ is given by $$\dim_H(A) = \inf\{s>0\...
0
votes
0answers
29 views

Showing that the packing dimension is countably stable

For a bounded set $F \subseteq \mathbb R^d$, let $\underline{\dim}_B F$ be the lower box-counting (or Minkowski-)dimension and $\overline{\dim}_B F$ the upper box-counting dimension, and let the lower ...
3
votes
1answer
116 views

Dimension of the graph of a product of continuous functions

Let $f,g\in{\mathcal C}[0,1]$ (continuous real-valued funtions defined on $[0,1]$). What can be said about the (Hausdorff or box-counting) dimension of the graph of $f\cdot g$ in terms of the ...
1
vote
1answer
56 views

Box dimension, graphs and sum of functions

Let $f,g:[0,1]\to{\mathbb R}$ be two continuous functions, Is it true that $dim_B(graph(f+g))\le \max\{\dim_B(graph(f)),dim_B(graph(g))\}$? It is well known (in Falconer book is an exercise) that ...
1
vote
1answer
32 views

Homotopy dimension of homotopy domination

The homotopy dimension of a space $X$ is the smallest covering dimension of any space homotopy equivalent to $X$. Assume that for topological spaces $X$ and $Y$, there exist maps $f:X\to Y$ and $g:...
0
votes
1answer
56 views

Does a space have dimension at most 1 if it contains no plane?

Let $X$ be a separable metric space. Suppose that $X$ contains no subspace homeomorphic to the plane. Is it necessarily true that the dimension of $X$ is $\leq 1$?
0
votes
0answers
9 views

The dimension of these matrix $\mathbf 0_{N_B}$,$\mathbf 0$ and $I_{N_B}$

I saw these formula in this paper: https://arxiv.org/pdf/1806.09814.pdf The (24a) formula is $\tilde {\mathbf H}^* + \mathbf Z_0^*=\mathbf 0_{N_B}$ and $\mathbf Z_0^*\mathbf W_B^*=\mathbf 0$,but as i ...
0
votes
0answers
31 views

Upper bound on Hausdorff dimension of set

I'm trying to solve the following question: Let $2\leq b\in \mathbb{N}$, $A\subseteq [0,1)$ and $\mu$ be a finite Borel measure such that $\mu(A)>0$. Denote $I_{n,b}^j:=\Big[ \frac{j-1}{b^n}, \...
2
votes
0answers
70 views

Dimension of a hypersurface of $\mathbb C^n$ / of a cut by a hypersurface

Defining the domain $$\Gamma[V]=\mathbb C [\bar x ]/I(V)$$ for any irreducible variety $V\subset \mathbb C^n$ (by variety, I mean only zero set of a family of polynomials), $\Gamma(V)$ for the field ...
2
votes
1answer
81 views

Exercise 1.8 from Hartshorne

(Hartshorne 1.8) Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \...
2
votes
1answer
61 views

Existence of dimension function (i.e., exact gauge function)

Motivation: Some sets have Hausdorff dimension $\alpha$ but have zero $\alpha$-dimensional Hausdorff measure. These sets may have another dimension function; i.e., a function $h:[0,\infty)\to[0,\infty)...
-3
votes
1answer
82 views

how to solve Gaussian integrals with three easy cases

I am using the the book called street mathematics to learn more about dimensional analysis. I am trying to understand a problem in the book. The question is to use dimensional analysis to find the ...
0
votes
1answer
35 views

What properties of $X$ guarantee that every open subset can be written as an increasing union of clopen sets

Let $X$ be a Cantor set. Then it is know that for any open subset $U$ of $X$, we can write $U$ as an increasing union of clopen subsets (in $X$). The same holds for any compact, metric, zero-...
0
votes
0answers
80 views

Calculate the dimension of Hilbert Curve

Is it possible to mathematically calculate the fractional dimension of Hilbert Curve using the formula D = log(N) / log(1/r) where D denotes the dimension of the ...
0
votes
0answers
37 views

How to define $\operatorname{dim}(\{0\})$ and $\operatorname{ht}(A)$?

Matsumura's "Commutative Algebra", Chapter 5, Page 72. It follows from the definition that $\operatorname{ht}(\mathfrak p)=\operatorname{dim}(A_{\mathfrak p})\quad (\mathfrak p\in \operatorname{...
1
vote
3answers
55 views

Why isn't a spherical surface three-dimensional?

According to the definition, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. A surface such as a ...
1
vote
0answers
25 views

Get approximate dimension of graph from adjacency matrix?

Imagine a huge graph that is planar apart from a few lines. (like mini wormholes). Is there a formula to get the approximate dimension of the graph from the adjacency matrix? My thoughts were to get ...