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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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
31 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
20 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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15 views

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
23 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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67 views

(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
28 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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2answers
123 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
86 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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32 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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62 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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2answers
38 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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1answer
36 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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0answers
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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0answers
37 views

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
66 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
51 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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1answer
49 views

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
39 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
64 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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12 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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47 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
187 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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663 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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22 views

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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1answer
64 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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60 views

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
32 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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34 views

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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52 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
28 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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50 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
47 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 ...
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13 views

Intersection of 2 continua with an arbitrary number of dimensions using matrices

If I have 2 continua (i.e. a line, plane, space, hyperspace etc.) of an arbitrary number of dimensions, each described parametrically as: x = xs + x1*t + x2*u + x3*v + ... + xn*k y = ys + y1*t + ...
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2answers
78 views

Is it possible to construct a smooth curve with fractional Hausdorff dimension?

It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true? If a curve has a fractional Hausdorff ...
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3answers
83 views

Where do extra dimensions in gradient come from?

The gradient of a scalar function $f\colon \mathbb{R}^n \to \mathbb{R}$ is a vector-valued function $\nabla f\colon \mathbb{R}^n \to \mathbb{R}^n$. Since applying a function can't increase ...
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1answer
66 views

rectangular paddock, dimensions, maximise area it encloses

Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot, ...
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1answer
152 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
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1answer
29 views

How many minimally sides are needed to fully enclose a volume in an $n$-dimensional spaece?

Imagine I have some unbendable*, but cuttable metal-plates. One side is silver, the other one is blue. Now I want to use them to build somekind of case (with no opening) that is fully blue. It just ...
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2answers
115 views

Box-Counting Dimension with finite resolution

Does the method of determining dimension of a shape via the Box-Counting dimension (Minkowski–Bouligand dimension) have to be performed on fractals (objects that look the same at all scales), or can ...
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28 views

Multi-dimensional naughts and crosses: victory for first player?

In a $2$-dimensional checkerboard which is $2\times2$, the player going first necessarily achieves $2$ in a row. In a $3$-dimensional board which is $3\times3\times3$, a player going first using a ...
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1answer
19 views

Maximal subset with finite Assouad-Nagata dimension

Given some space $X$ with non-finite Assouad-Nagata dimension. Is it possible for a subset $Y \subset X$ with finite Assouad-Nagata dimension to exist such that $Y$ is maximal in the sense that if any ...
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36 views

Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in ...
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112 views

When a function is a dimension?

The concept of dimension is used in many different contexts. Generally a dimension is a function that has as domain some family of sets ad has value on a set that, in the most common situations, is ...
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1answer
31 views

What is a family of lines?

And furthermore, how do we define the dimension of the family? My question derives from the stating of the Wolff axiom - Let $\mathcal{L}$ be a two-dimensional family of lines in $\mathbb{R}^3$ such ...
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1answer
46 views

Diameter of a 10-ball in a 10-box is larger than the side length of box?

I came across this idea in a lecture on elementary topology. While it makes sense algebraically, I'm hoping someone could shed some light on the way this is possible. So you begin with a square of ...
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1answer
47 views

Maximal Distance of a graph

So I have the following graph $G = (V, E)$ with $V = [d]^n$ and $E = \{\{(a_1,\dots , a_n), (b_1, \dots , b_n)\} | a_2 = b_1 a_3 = b_2, \dots a_n = b_{n-1}\}$. That is called a $(n, d)$-dimensional ...
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1answer
63 views

Dimension of a certain $L^p$ quotient space.

Define $L^p_0 := \{ f \in L^p : \int f = 0 \}$. I am trying to calculate the dimension of the cokernel of the inclusion operator $i:L^p_0 \to L^p$. That is, I am trying to calculate $$\dim ( L^p / ...
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90 views

Lower Bound of Hausdorff Dimension of Cantor Set

Consider a Cantor set $E$ where the intervals at every level of the construction maintain a minimum spacing and have a finite number of intervals on each level. I have two questions regarding finding ...
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100 views

How to find the number of spanning trees for a cube?

Can you tell me a way of finding the total number of spanning trees in a $Q_d$ undirected labelled graph for $d = 3$. I know that the answer is 384, but the way (I know there are many.) of finding ...