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.

learn more… | top users | synonyms

0
votes
1answer
15 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
8
votes
1answer
280 views

When is a function 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 ...
0
votes
2answers
542 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 ...
3
votes
1answer
105 views

Is a hypersphere of non-integer dimension a fractal?

Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$ S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ allows to calculate the surface of a ...
10
votes
2answers
2k views

Can a polygon be one dimensional?

When looking up the definition of polygon, Wikipedia tells me: In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing ...
0
votes
0answers
10 views

Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
1
vote
1answer
352 views

Hausdorff Dimension of a Smooth Manifold

I read a book about fractals which stated without proof: Every $m$-dimensional $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove this?
1
vote
0answers
15 views

When the self-similar dimension and the Hausdorff dimension are different?

By the en.wikipedia, for the self-similar sets in a metric space, the self-similar dimension and the Hausdorff dimension are often the same, but not always. Is there a known sufficient-necessary ...
0
votes
0answers
13 views

Simulating 4D on a 3D mean

Why if we can simulate 3D on a 2D mean, why isn't possible to simulate 4D on a 3D mean (real world)? Has someone tried?
4
votes
2answers
206 views

How to list all possible dimension of $\ker{T},\ker{T^2},…,\ker{T^{k-1}}$ and the corresponding canonical forms?

Let $V$ be $5$-dimension vectorspace, and $T:\ V\rightarrow V$ a nilpotent linear transofrmation of order (index) $k$ where $1\le k\le 5$. How to list all possible dimension of ...
1
vote
1answer
29 views

System of parameters in Noetherian local rings

I'm trying to understand the theorem for systems of parameters in Noetherian local rings, which says: Let $R$ be a Noetherian local ring with maximal ideal $m$. Then there exists an $m$-primary ideal ...
2
votes
0answers
45 views

$\mathrm{R}^n$ for non-integer values of $n$?

If I have a set of all possible $x$ for $x \in \mathrm{R}^n$ for non-natural number $n$, what meaning would this have and how would it be used? For instance, if I had $\mathrm{R}^{2.5}$, would this ...
1
vote
3answers
35 views

Extension of Up/Down/Right/Left in n dimensions

I'm wondering if there are extensions of the right/left/up/down concepts in n dimensions, i.e. "if $x_0$ points ahead of me, I'm looking LEFT/RIGHT along $x_1$, UP/DOWN along $x_2$, XXX/YYY along ...
1
vote
0answers
38 views

Base change and relative dimension

Let $B$ a ring of Krull dimension $1$ and suppose that $\text{Spec }B$ is irreducible. And let $f:X\to\text{Spec}(B)$ a $B$ scheme with the following properties: $X$ is projective, regular and ...
0
votes
1answer
152 views

Conditions so that Lebesgue Covering Dimension and “Usual” Dimension are Equal

The definition of covering dimension is as follows: The ply of a cover is the smallest number $n$ (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement ...
2
votes
2answers
231 views

The 1-dimensional Hausdorff measure of a curve in the plane

For a set $X\subseteq\mathbb{R}^2$, let $H^1(X)$ be its 1-dimensional Hausdorff measure. Suppose $X$ is a regular curve (say, a graph of a continuous function $f:\mathbb{R}\to\mathbb{R}$). In that ...
5
votes
1answer
87 views

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
1
vote
0answers
16 views

Bounding below Hausdorff measure of conected set

I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$. In the answer there, it was suggested to use a Lipschitz ...
2
votes
1answer
40 views

Dimension about space of matrices of order 3 over the field of the real.

Consider the vector space of the matrices of order 3 over the field of the real $M_{3}\left(\Re\right)$ numbers. and let S be the subspace such that is spanned by the matrices of the form $AB-BA$. ...
7
votes
1answer
219 views

What would a tesseract actually look like?

Say our world is actually 4-dimensional, and I have a tesseract sitting on my table. To a being only capable of perceiving 3 dimensions, what would it actually look like? Would it just look like a ...
0
votes
1answer
37 views

Is the dimension $-1$ the real $0$th dimension and does this all make sense?

I know there are at least two questions on this site that ask about the negative dimensions. But I want to ask something more than that. We have a number line. It contains all the real numbers we can ...
0
votes
1answer
29 views

Relation between eigenvalue and the kernel

Consider the $0$ matrix. It has one eigenvalue: $0$, and the dimension of its eigenspace is $n$, since it sends everything to $0$. If I have $n$ distinct eigenvalues, one of them zero, then the ...
1
vote
2answers
238 views

How to correctly calculate the fractal dimension of a finite set of points?

The box-counting dimension is defined by: $\lim\limits_{\epsilon \to 0} \dfrac{N(\epsilon)}{1/ \epsilon}$ What works well if you are solving algebraically or if you can recursively generate more ...
0
votes
1answer
21 views

Lebesgue measure of union of a finite number of hyperplane and isolated points in $\mathbb{R}^n$

Here a question for you guys. Let $S_1,S_2,\dots, S_l\subset \mathbb{R}^{n}$ be some affine spaces of dimension $n-1$, and let $P_1,P_2,\dots,P_j\in\mathbb{R}^n$ be some points. Let ...
3
votes
6answers
6k views

Orthogonal planes in n-dimensions

In three dimensions two planes are orthogonal when their normal vectors are orthogonal (their inner product is zero). For example, planes $xy$ and $xz$ are orthogonal because the vectors $\hat{z}$ and ...
4
votes
1answer
232 views

How can I understand is the picture $2D$ or $3D$

I can not understand is this picture 2D or 3D. What is the rule or condition to be a 2D or 3D picture. How can I understand that? Please help me!
308
votes
0answers
15k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
1
vote
1answer
29 views

Polish groups having finite covering dimension

The paradigm examples of Polish groups that have finite covering dimension are obviously $\mathbb{R}^n$ for any finite $n$. They are also locally compact. Is it possible to construct a sequence $G_n$ ...
1
vote
1answer
52 views

Are there any practical applications for higher dimensional geometry?

My current understanding of higher dimensions is that we can take geometric properties from 2D or 3D, and simply extend them further theoretically. So in the same way that 3D objects can be projected ...
0
votes
1answer
14 views

Dimensionality of shapes in two space

I'm having trouble properly understanding the dimension of shapes. Take a square or triangle. Are the boundaries of these two-dimensional shapes one-dimensional? Is the entire figure two dimensional? ...
4
votes
2answers
101 views

Why study dimensions?

I am quite new to the forum so please feel free to correct/give pointers if I am posting something in the wrong place. I have been perusing "Dimension Theory" by Witold Hurewicz & Henry Wallman ...
1
vote
2answers
57 views

Dimension of a Noetherian topological spaces

We know that the definition of (Krull) dimension of a Noetherian topological spaces $X$ is the following: $$ \dim X=\max\{n\in\mathbb{N}\mid\emptyset=Z_{-1}\subsetneqq Z_0\subsetneqq\dots\subsetneqq ...
0
votes
1answer
26 views

Proving rank$(S\circ T)\le$ rank $S$ and null$(S\circ T)\ge$ null $T$ [closed]

Let $T$:$V\rightarrow W$ and $S$:$W\rightarrow U$ be linear transformations. I need to prove the two statements in the title. I don't know how to approach this problem. I am quite sure it involves ...
0
votes
1answer
36 views

Using Rank-Nullity Theorem

If $T$:$V\rightarrow W$ and $S$:$W\rightarrow U$ are linear transformations: How do we define $\text{rank}(ST)$ and $\text{null}(ST)$? I know $\text{rank}(S)=\text{dim}(\text{Im}(S))$ and ...
1
vote
3answers
61 views

Proving that the space $\mathbb{P}$ of all polynomials is not finite dimensional

Just want to know if my approach to this problem is correct: Suppose that the space P of all polynomials is finite dimensional. Then we can call it $\mathbb{P_{n}}$ with $n<\infty$. Therefore the ...
0
votes
0answers
22 views

Can you characterize dimension in terms of the touching axioms?

One alternative axiomatization of topology is the touching axioms. It's basically the same as the closure axioms, and is in fact equivalent to the open set definition. Can we define dimension in a ...
0
votes
2answers
56 views

Let $S:U\rightarrow V, \ T:V\rightarrow W$ and if $S$ and $T$ are both injective/surjective, is $TS$ injective/surjective?

For injective side, in my opinion, we can find two square matrices $A, B$ with each injective but $AB$ has a zero row ($\dim\ker(AB)\ne 0$), so $TS$ is not injective. But I don't know how to find such ...
30
votes
4answers
1k views

How to tell what dimension an object is?

I was reading about dimensions and in the Wikipedia article it states the following: In mathematics, the dimension of an object is an intrinsic property independent of the space in which the ...
4
votes
2answers
124 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 ...
1
vote
1answer
28 views

Is range of completely continuous of bounded set finite dimensional set?

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Let $\Omega$ is a bounded set of ...
3
votes
0answers
65 views

Proving how to reduce a Brownian walk on a plane to a line (2D to 1D)

I have a Brownian motion on a plane and would like to find the time of when it is expected to hit a set of parallel lines, i.e the hitting time. In order to do so, I understand that I can reduce the ...
1
vote
0answers
35 views

Compute Hausdorff dimension of cantor set.

I have to show that $0<\mathcal H_{\alpha }^\delta(\mathcal C)<\infty $ where $\mathcal C$ is the cantor set, $\alpha =\frac{\ln(2)}{\ln(3)}$ and $$\mathcal H_\alpha ...
1
vote
4answers
35 views

How to show $T$ is bijective in the following condition?

Let $T:\ V\rightarrow W$ be a linear transformation, if $\dim(V)=\dim(W)$, $\{v_1,...,v_n\}$ is a basis for $V$ and $\{w_1,...,w_n\}$ is a basis for $W$. Let $T:V\rightarrow W$ be a linear ...
2
votes
1answer
22 views

Dimension of the quotient space $\frac{C_{0}}{M}.$

Let $C_{0}=\{(x_{n}):x_{n}\in\mathbb{R},x_{n}\rightarrow 0\}$ and $M=\{(x_{n})\in C_{0}:x_{1}+x_{2}+\cdot\cdot\cdot+x_{10}=0\}.$ I have to find dimension of the space $\frac{C_{0}}{M}.$ According to ...
6
votes
1answer
64 views

Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?

This is not a very well-defined question. Are there any standard constructions of metric spaces, parameterized by real-valued $n \ge 1$, such that: When $n$ is an integer, the metric space is ...
0
votes
1answer
44 views

How to show $\{T(v_1),…,T(v_n)\}$ is a basis for $W$ if $T$ is bijective and $\{v_1,…,v_n\}$a basis for $V$?

Let $T:\ V\rightarrow W$ be a linear transformation, if $T$ is bijective and $\{v_1,...,v_n\}$is a basis for $V$, how to show $\{T(v_1),...,T(v_n)\}$ is a basis for $W$? Here is my thinking process: ...
0
votes
0answers
29 views

Define angle between “3D space” in 4D space

We can define the angle between the line in 2D space, and also can define angle between planes in 3D space, and is it possible to define the angle between 3D space in higher dimensions? And if it's ...
0
votes
0answers
14 views

Is it possible to think solid as the orthogonal projection from 4D?

Get an orthogonal projection's area from this fomular $S=S^{\prime}\cos{\theta}$ when the angle between two planes is $\theta$, is a popular idea, so I wonder is it possible to get a volume of various ...
1
vote
0answers
35 views

How to find family of functions

I want to find the family of functions with this property: $$\epsilon F\left(\frac{x}{\epsilon}\right)=\epsilon^r F(x)\quad\text{with}\ x,\epsilon, r \in \mathbb{R}$$ ($\epsilon$ and r are ...
1
vote
1answer
63 views

How to add and multiply on fractional vector space

How to add and multiply on fractional vector space Please, answer in layman terms. I don’t understand the notation of Supersimetry and Super vector spaces. If a fractional “vector space” (or his ...