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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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 ...
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$\mathrm{R}^n$ 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 ...
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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 ...
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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 ...
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139 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 ...
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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 ...
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70 views
+50

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 $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
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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 ...
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1answer
37 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 $AB-BA$. ...
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206 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 ...
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35 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 ...
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25 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 ...
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2answers
473 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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2answers
233 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 ...
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1answer
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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 ...
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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 ...
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231 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!
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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 ...
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1answer
28 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 $G_n$ ...
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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 ...
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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? ...
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179 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 ...
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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 ...
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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 ...
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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 ...
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35 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 ...
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3answers
60 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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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63 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 ...
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34 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 ...
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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 ...
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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 ...
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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 ...
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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: ...
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28 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 ...
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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 ...
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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 ...
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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 ...
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1answer
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How can one do dimensional analysis when units are not known?

In the sciences, we can do dimensional analysis and unit checks to verify whether or not the LHS and the RHS have the same units. If we have the following function:$$y=f(x)=x^{2}$$ what ensures the ...
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44 views

What is the Hausdorff dimension of $\mathbb{R}^d$

the Hausdorff dimension $\dim(A)$ of a Set $A \subseteq \mathbb{R}^d$ is defined as the unique Number $\alpha$ such that the $\beta$-dimensional Hausdorff measure $H_\beta(A)=0$ for $\beta>\alpha$ ...
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The critical value of an infinte sum over [0,1).

Consider $f(s)=\displaystyle\sum_{i=1}^\infty r_i^s$ where $s\in[0,\infty)$ and $0<r_i<1$. Under what conditions can we claim that there exists some $s$ such that $f(s)=1$. I know that some ...
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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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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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221 views

Is the Hausdorff dimension less than the box counting dimension?

I have been asked to prove that for a bounded set $F\subset\mathbb{R}^n$, $\dim_H F\le \underline{\dim}_B F \le \overline{\dim}_B F$ where $\dim_H F$ is the Hausdorff dimension, ...
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166 views

Poles of formal power series (Hilbert-Poincaré series)

How are poles and orders of poles of formal power series defined? The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, ...
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246 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq ...