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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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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+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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3answers
33 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 ...
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1answer
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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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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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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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 ...
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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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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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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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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 ...
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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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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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4answers
33 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 ...
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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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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 ...
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43 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: ...
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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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50 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 ...
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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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1answer
60 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 ...
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1answer
33 views

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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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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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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1answer
29 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
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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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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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47 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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19 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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25 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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47 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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$W$ 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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33 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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38 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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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
27 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!