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. In commutative algebra, dimension can be defined for commutative rings.

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Obtaining a solution space to an inverse kinematics problem (mapping a higher dimensional space onto a lower one)

I have a 15 dimensional space (corresponding to 15 joints of a robot, 5 joints for each of two legs, and another 5 joints for the right arm). I'll call this space A. Bear in mind that each of the 15 ...
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2answers
250 views

How to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$ based on the following assumption?

Suppose $U$ and $W$ are subspaces of a vector space $V$ such that $\dim(U) =\dim(W)$ and $U\ne W$, how to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$? My approach is to use ...
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Dimension of a space VS dimension of a function in this space [closed]

A colleague and I ran into a problem when we realised that we had a complete different understanding of dimensions. If we consider this function: $z(x,y)=x^2 + y^2$ Person A believes this function ...
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1answer
27 views

Calculating a Hausdorff Dimension from Formula

I am having difficulty computing the following formula for the Hausdorff Dimension for a type of Moran set called a partial homogenous Cantor set. ...
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1answer
60 views

Exercise 6 in section 50 in Topology textbook of Munkres.

I am looking for a reference of the proof of the following thoerem in Munkres it's exericse 6 in section 50 page 315 and it goes as follows: Let $X$ be a locally compact Hausdorff space with a ...
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1answer
15 views

opposite points in n-dimensional shapes

does anyone know if there is a generalized approach to generating opposite points within n-shapes such as an n-cube or n-sphere. I am trying to find out if a uniform distribution will be better at ...
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1answer
36 views

The dimension of the space of continuous functions that are piecewise polynomials of degree $k$

I am trying to calculate $$ \dim( \{ f\in C^0 ([a,b]) : f_{|[x_{j-1},x_j]} \in \mathcal{P}_k, j = 1,...,m \}) \text{ with }m,k \in \mathbb{N} $$ Is $m(k+1)$ correct? My thoughts: I have $m$ ...
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1answer
18 views

Diophantine approximation and covers

Suppose $\alpha > 2$. Let F be the set of real numbers $x \in [0,1]$ for which the inequality $||qx|| \le q^{1-\alpha}$ is satisified by infinitely many positive integers q. For each q, let $G_q$ ...
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24 views

Dimension of vector spaces involving differential equation

Let V denotes the vector space $C^5[a,b]$ over $\mathbb R $, and $W=\{f\in V:\frac{d^4f}{dt^4}+2\frac{d^2f}{dt^2}-f=0\}$.Then $dim(V)=\infty$ and $dim(W)=\infty$ $dim(V)=\infty$ and $dim(W)=4$ ...
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40 views

Codimension and height of prime ideals

Definition: Let $Z$ be an irreducible closed subset of $X$. Then the codimension $\textrm{codim} (Z,X)$ is the supremum of integers $n$ such that there exists a chain $$ Z = Z_0 < Z_1 < ...
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32 views

$W$ intersection of $(n-1)$ dimensional subspaces

I have got a good (I think so) intuition of this problem but I am not being able to write down the crucial steps correctly. Let $V$ be a $n$ dimensional vector space over field $F$ . Let $W$ be a ...
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1answer
26 views

Hausdorff dimension calculation related to Jarnik's theorem

Let $$F=\{x \in R:||qx||\le2q^{1-\alpha}\log q \text{ for infinitely many } q \in \mathbb{R}\}$$ Show for $\alpha>2$, $\dim_H F\le 2/\alpha$. Jarnik's theorem (By Falconer) says: Suppose ...
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44 views

Hausdorff dimension calculation of union of sets

$F$ is a Cantor set in $(0,1)$, $\dim_HF=1/5$. What's the $\dim_HE$ where $E=(F×R)\cup(R×F)$? By the product properties, I know that and $\dim_H(F×[0,1])=6/5=1+1/5$, which is the sum of hausdorff ...
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1answer
31 views

Hausdorff dimension of F and f(F)

We have F being a subset of R, [-1,1], while f:R->R, where f(x)=x^2. What's the Hausdorff dimension of F and f(F)? I think the dim(F)=2(length) and dim(f(F))=1, is it correct? Thanks,
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39 views

Sets with Hausdorff-Measure 0

The $\alpha$-Dimensional Hausdorff-Measure of a Set A is defined as $H^\alpha (A)=\inf_{A\text{ is countable covering}}\sum_{A'\in A} diam(A')^\alpha$. It is easy to show, that for every set ...
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1answer
20 views

Hausdorff dimension of a countable set

I don't understand why the Hausdorff dimension of a countable set in $\mathbb{R}^n$ is $0$. Can someone please give me a hint? Thank you!
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31 views

Intersection/conjunction matrix and transforms?

I'm looking for a method to understand particular types of inclusion/intersection between objects (sets). Say I have a collection of sets, and between every pair of sets [A,B] I can derive a number ...
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2answers
27 views

Making equations dimensionless

I have a equation of motion for a forced pendulum show below $$ {d^2\theta\over dt^2} = -{g\over L}\sin\theta + C\cos\theta\sin(Dt) $$ $L=10$ cm, $C=2\ \hbox{s}^{-2}$ and $D=5\ \hbox{s}^{-1}$. I am ...
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1answer
13 views

Finding the similarity dimension of a variation of the Cantor Set.

If we take the Cantor set and instead of removing the interval $[1/3, 2/3]$, we remove the open interval $[x,1-x]$, with $0<x<1/2$, will the similarity dimension change? What I think is that we ...
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Is there a simple proof that the covering dimension of the complement of $Q^2$ is one?

It can be shown that dim $R^2\setminus Q^2$ is 1 since this set has no interior. Is there a simple direct proof that does not use the equivalence of the small inductive dimension with the covering ...
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1answer
18 views

Is a 2D object on a plane that gets curved still considered 2D or is it now 3D?

Imagine a 2D square drawn on a flat membrane. The square has length and width, but no depth (obviously; it's a square). Now take said membrane and place it with the middle of the square centered atop ...
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(quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of ...
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1answer
24 views

Showing an equation satisfies laplace equation

I'm completely lost with the laplace equation I've searched different explanations of it on google and on this website and nothing is helping explain it. The question I was given is: Show that the ...
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39 views

Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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1answer
73 views

Continuous one-to-one mapping from a subset $K \subset \mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f\colon \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a continuous function and $K\subset \mathbb{R}^n$ a subset of positive Lebesgue measure. Is it possible that $f$ is one-to-one on $K$? If $K$ ...
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28 views

How to characterize the dimension of a manifold using homology?

This might be a trivial question but I'm a physicist, not a mathematician. For me, the n dimensional euclidean space is n dimensional as a vector space. I have heard however that there more intrinsic ...
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58 views

Singular points of one-to-one mapping on rectifiable set

Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is: Can there exist a one-to-one Lipschitz function ...
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24 views

Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...
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32 views

Example of equivalent linear transformations with different rank

As stated in title. When it comes to infinite dimensional vector spaces I guess I just lose track. Can somebody give me an example of equivalent linear transformations with different rank? [EDIT]: ...
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9 views

Codimension of subspaces of a sphere

Let consider the three following sets : $S_1 = \{ (x,y,z) \in \mathbb{R}^3\; |\; x^2+y^2+z^2=1 \}$ $S_2 = \{ (x,y) \in \mathbb{R}^2\; |\; \exists z\in \mathbb{R}, \; x^2+y^2+z^2=1 \}$ $S_3 = \{ x ...
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Interpreting infinite integer lattice as a manifold of negative dimension

Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the ...
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1answer
42 views

Definition of Minkowski dimension

I'm trying to understand the definition of Minkowski dimension given by Wikipedia here. In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting ...
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1answer
45 views

Hausdorff measure of a subset of a smooth manifold

Assume that $X$ is a subset of a compact smooth manifold. We know if $\mathrm{dim}_H(X)=d$, then $m_{d'}(X)=\infty$ for $d'<d$, and $m_{d'}(X)=0$ for $d'>d$. But we have no information about ...
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Where in the theory of higher dimensions do Bernoulli numbers arise? [closed]

In the theory of metric spaces of higher dimensions, where the Bernoulli numbers arise? Are there any formulas for instance for volumes of manifolds involving Bernoulli numbers?
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An open covering of the Topologist's sine curve.

How does an open covering of of the topologist's sine curve look like? I am asking since I want to show that it has a topological dimension 1.
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1answer
31 views

Finding dimension of a submodule

Let $G= (\mathbb{C}^3, A)$ be the $\mathbb C[x]$-module given by $$ A=\left( \begin{matrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ ...
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1answer
44 views

Associated graded ring isomorphic to polynomial ring implies regularity

Let $(A, \mathfrak{m}, k)$ be a Noetherian local ring of dimension $d$. I would like to prove, or rather understand, why the following holds: If the associated graded ring ...
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9 views

Codimension 1 - points on curve

Let C be a plane curve. What are the objets of codimension 1 in C ? Are points of codimension 1 on C ? Thanks
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37 views

Using the dimension formula to prove isomorphism

Let V be a finite-dimensional vector space and $T: V\rightarrow V $. T is a linear transformation. Use the dimension formula to prove that if T is injective, it must also be surjective; if T is ...
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1answer
143 views

Vector spaces with fractional dimension

Can the notion of vector space or algebra over a field be meaningfully extended to fractional dimensions, so that for example $\mathbb{R}^{-2/3}$ makes sense? Has this been explored somewhere? I know ...
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2answers
644 views

Is zero the center of the numeric sequence?

The numeric sequence has symmetry on zero, with equal infinities of cancelling out + and - values on either side. Can numbers be said to have different centers of symmetry than zero? Is it possible ...
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How to determine if two points with n dimensions each are less than or greater than one another

I am writing a program that needs to compute if a point with an arbitrary amount of dimensions is less than or greater than another point with an arbitrary amount of dimensions. It is easy to ...
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63 views

Basis and dimensions

How do i find the a basis and dimension for $A[x]$? Consider the subset of $R[x]$ given by $A[x]:=\{q(x)$ element of $\mathbb R_4[x]$ such that $q(2)=0=q(-3)\}$ I'm a bit confused because there are ...
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Rank of a matrix over a principal ideal domain

I apologize if my question is stupid but I'm not very familiar with matrices over a principal ideal domain $R$ (For example, $R=\mathbb{Z}$ or $R=\mathbb{R}[X]$). I was wondering how to define the ...
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1answer
28 views

Codimension of a sphere

Let $\cal{S}$ be the n-sphere as a n-dimensionnal submanifold of $R^{n+1}$. The codimension of $\cal{S}$ in $R^{n+1}$ is $1$ right ?
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Why is there a need for other coordinate systems?

I have been wondering a lot about this. I have just reached A-Levels, and I have chosen the course Further Math. In class, we were talking about the Parallel coordinate system, but I didn't really see ...
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40 views

Latitude/Longitude Dimension Reduction - 2D to 1D and back again

I have a 2D array (latitude, longitude) which I would like to convert/map to a 1D integer (i.e. for machine learning regression purposes). I would then like be ...
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1answer
27 views

Let $V_1,V_2$ be subspaces of $V$. If $\dim(V_1+V_2)=\dim(V_1 \cap V_2) + 1$ then prove that $ V1 \subseteq V2 $ or $V2 \subseteq V1$.

We know that $\dim(V_1+V_2)=\dim(V_1)+\dim(V_2)-\dim(V_1 \cap V_2)$, But $\dim(V_1+V2)=\dim(V1 \cap V2)+ 1$ (given), Now if we assume that $V_1 \subseteq V_2$ , $V_1 \cap V_2= V_1$, Then at one side ...
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2answers
38 views

Hausdorff dimension of graph of function

This question came up on an exam Decide the Hausdorff dimension of the graph of the following function for $x>0$ $$y = \log(1+x)\sin\frac{1}{x}$$ In the course, we only touched upon the subject ...
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2answers
43 views

Does finite covering dimension imply local compactness?

I have a space which is not locally compact and I'm trying to see if I can say anything about the dimension of the space. I suspect that it is not finite dimensional but I have thus far been unable to ...