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

2
votes
1answer
360 views

Best way to plot a 4 dimensional meshgrid

I have $4$ variables $X$, $Y$, $Z$ and $C$, and I want to plot these on a graph. Usually I would just plot the surface $X$, $Y$, $Z$ and then use color to represent the $4$th dimension, as shown ...
1
vote
0answers
19 views

On the definition of (small) inductive dimension

A regular topological space $X$ has inductive dimension smaller or equal to n if and only if: ($n=-1$) $X=\emptyset$; ($n>-1$) The space $X$ has a base of opens $\mathscr{B}$ such that, for all ...
0
votes
1answer
88 views

Hausdorff content and Hausdorff measure

I am dealing with the Hausdorff dimension and I came across two different ways of defining this dimension. This question is possibly related to Hausdorff Measure and Hausdorff Dimension but the ...
1
vote
1answer
34 views

Question on density of a set

I am reading a paper on complex dynamics and Hausdorff dimension, and there is a result that I can't prove. I have the following situation. For each $k=1,2,...$, we denote $E_k$ a finite collection ...
1
vote
0answers
35 views

Number of touching triplets in hypersphere packing contact graph

I would like to know what is curently known about the number $N$ of touching-triplets involving a given vertex $V$ in the contact-graph of $d$-hypersphere packings. For $d=2$ and $d=3$, one has $N = ...
4
votes
1answer
106 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?
1
vote
2answers
84 views

How to imagine a fractal dimension?

I am interested in fractal dimensions and more or less familiar with the self similarity, boxcounting and Hausdorff dimension. If someone asked me “What is a fractal dimension?”, my answer would be ...
3
votes
2answers
241 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
3
votes
1answer
222 views

Cantor set exercise

This is an exercise from Abbott's real analysis book. It's exercise 3.4.4.(b) on page 93. I couldn't find a definition of ''dimension'' in the book. The only thing I could find is something on page ...
1
vote
0answers
15 views

Can correlation dimension of an attractor exceed the dimension of the space?

Here is the definition of the correlation dimension: http://en.wikipedia.org/wiki/Correlation_dimension Is there a proof that the correlation dimension cannon exceed the dimension of the space?
3
votes
1answer
70 views

Principal ideal domain is universally catenary …

... actually, even more general statement is true: Theorem. Every regular ring is universally catenary. (see for example Algebraic Geometry by Qing Liu, Corollary 2.16, Chapter 8) Though, the ...
0
votes
1answer
103 views

Hausdorff dimensions of smooth but non-rectifiable curves

Smooth curves of finite length have a Hausdorff dimension of 1. How about smooth but non-rectifiable (i.e. infinite-length) curves? Are they also of Hausdorff dimension 1, or does it depend on the ...
2
votes
1answer
86 views

Behavior of Hausdorff dimension under homeomorphisms

Let $X$ and $Y$ be metric spaces, $f : X\rightarrow Y$ a homeomorphism. Denote by $\dim_{\mathcal H}$ the Hausdorff dimension. I know that it is possible that $\dim_{\mathcal H} Y < \dim_{\mathcal ...
0
votes
1answer
247 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
0
votes
1answer
23 views

the dimension problem of complex projection

Is it true that $\operatorname{dim }H^{0}(P^{n},P(T^{*}P^{n}))>0$? That is, is there a global holomorphic section? Here $P^{n}$ is $n$-dimensional complex projection space and ...
7
votes
2answers
232 views

What does it mean to be $0.9-$Dimension?

We can visualize $1\mathrm D$, $2\mathrm D$, $3\mathrm D$ and we can think of a higher $M-\mathrm {Dimension}$ where $M\in \mathbb N^+$as a vector. But I recently learned that there are non integer ...
1
vote
0answers
57 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
0
votes
1answer
77 views

Embeddings and intersection of clopen subsets

My new question is again in the context of Hausdorff 0-dimensional spaces. We say that S subspace of a space X is a 2-embedding if for every continuous function with domain S and codomain 2(the ...
5
votes
2answers
99 views

Question about “equivalent” definitions for small inductive dimension of topological spaces

$\DeclareMathOperator{\ind}{ind}$I've been reading through this document, trying to get a better handle on the interrelationships between various notions of topological dimension, and I came across ...
0
votes
1answer
46 views

Derricks Theorem for D= 2 and 3

According to Derrick's theorem we can write \begin{align} E &= \frac{1}{2} \int d^Dx \frac{1}{\lambda^2}\left( \nabla \phi_i (\frac{x}{\lambda})\right)^2 + \int d^Dx ...
5
votes
2answers
131 views

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalance does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
8
votes
4answers
159 views

Dimension of an object?

One simple way to define dimension is "the number of numbers required to describe an object." If we consider the set of circles, we can describe each of them by one number -- radius, or ...
6
votes
2answers
212 views

Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip?

Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity ...
0
votes
1answer
26 views

stepping on N-d hypercube

Consider a N-d hypercube. All the nodes (cube corners) are connected with arches, and there are 2^N corners. If I begin at a specific node, to how many nodes can I reach if I make r moves along r ...
1
vote
0answers
38 views

Reference Needed: Krull Dimension equals Minkowski Dimension

I'm looking for a reference that basically states that the Krull dimension of an algebraic set in $\mathbb{R}$ equals the Minkowski dimension of the set. It is common knowledge that the Krull ...
2
votes
2answers
98 views

Space of global sections for a smooth projective curve of genus $g$

Let $X$ be a smooth projective curve of genus $g$, $T_{X}$ its tangent bundle and $H^{0}(X,T_{X})$ the space of global sections for $X$. What is $\dim H^{0}(X,T_{X})$ and why?
0
votes
1answer
99 views

Understanding Sierpinski carpet formally

In this paper, one definition of carpet(Sierpinski) is given as follows: A metrizable topological space is a carpet iff it is a planar continuum of topological dimension 1 that is locally connected ...
2
votes
1answer
112 views

Two Definitions of Minkowski Dimension

I'm currently reading a paper. Let $F\subset\mathbb R^n$ and $\epsilon\gt0$, the paper defined $m^s(F):=\liminf_{\epsilon \to 0}\epsilon^{s-n}\lambda(F_\epsilon)$ and $M^s(F):=\limsup_{\epsilon \to ...
0
votes
2answers
200 views

Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
3
votes
1answer
203 views

Proof details of Theorem 11.1 in Atiyah-Macdonald

I have some trouble filling in the details of this proof from Atiyah-Macdonald. In this result, the authors assume what follows: 1) $A = \oplus_{n=0}^\infty A_n$ is a Noetherian graded ring, and ...
4
votes
2answers
276 views

Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere ...
4
votes
3answers
100 views

Converse of a dimension lemma

Consider the following lemma. It comes from the Stacks Project. Lemma 9.59.11. Suppose that $R$ is a Noetherian local ring and $x\in\mathfrak m$ an element of its maximal ideal. Then $\dim R\le ...
1
vote
2answers
192 views

Dimension theory “based on $\mathbb R^n$”

This question is somewhat vague, so please be gentle with me. I want to know if there is some definition of topological dimension that has $\mathbb R^n$ as a "paradigm", something like 'A nice ...
2
votes
1answer
113 views

Finding Potential with d dimensions terms

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
3
votes
1answer
128 views

Krull dimension of a $\mathbb Q$-algebra

I'm trying to find the Krull dimension of $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$. My professor said that I have to consider that $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$ is a $\mathbb{Q}$-algebra but I ...
4
votes
3answers
768 views

Do negative dimensions make sense? [duplicate]

Some time ago I read in a popular physics book that in M-theory, there are some "things" which can be said to have dimension $-1$. Probably, the author was vastly exaggerating, but this left me ...
1
vote
0answers
33 views

How can i evaluate a parabolic region in 2-Dimensions onto n-Dimensions?

I am working on the K-Nearest Neighbor algorithm and have the solution to a problem as a parabola in 2-dimensional space. Should I continue this solution onwards to n-dimensions (and thereby create ...
4
votes
1answer
117 views

Is there a dimension which extends to negative or even irrational numbers?

Just elaborating on the question: We all use to natural numbers as dimensions: 1 stands for a length, 2 for area, 3 for volume and so on. Hausdorff–Besicovitch extends dimensions to any positive real ...
2
votes
1answer
88 views

Radial coordinate evaluation

Details of the question can be found in the article equation(55,56) A radial coordinate $R$ defined by \begin{equation} r=\frac{2R}{\kappa(1-R^2)} \,, \end{equation} where $\kappa$ is a constantand ...
4
votes
2answers
65 views

What is the name for the region enclosed by an $n$ dimensional object?

In a $1$ dimensional object, the name for the region enclosed by it is the length of the object. In a $2$ dimensional object, the name for the region is the area of the object. In a $3$ dimensional ...
1
vote
2answers
60 views

spherically symmetric configurations

$$\Delta S -S +S^3=0$$ How this Differential equation can be written in this form: \begin{equation} \frac{d^2S}{d\rho^2}+\frac{D-1}{\rho}\,\frac{dS}{d\rho} -S+S^3=0 \end{equation} Which is ...
-1
votes
3answers
129 views

A question about direct sums of subspaces…

Let $W_1$ be a subspace of a finite dimensional vector space $V$. Prove that there exists a subspace $W_2\subset V$ such that $V=W_1\oplus W_2$. EDIT$^1$: This may be of use here: What ...
10
votes
1answer
171 views

Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
2
votes
0answers
75 views

Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring ...
1
vote
1answer
120 views

Reverse Hölder Continuity and Hausdorff dimension

Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
1
vote
2answers
52 views

Odd valued dimensional number impossible to build?

Using numbers of the form $$\alpha_1+\alpha_2e_1+\alpha_3e_2+...+\alpha_ne_{n-1}$$ where $\alpha_n\in\Bbb R$ and for all $a≠b, \alpha_a≠\alpha_b$ with $e_n^2=-1$, can these numbers exist for an odd n? ...
4
votes
3answers
137 views

$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$

How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
0
votes
0answers
78 views

How to show an ideal is zero-dimensional? [duplicate]

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional. How do I go about showing this?
0
votes
1answer
48 views

Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
0
votes
1answer
33 views

$\sum^{k}_{i=1} \dim(V_i) > n(k-1) \implies n>\sum_{i=1}^k\bigl(n-\dim V_i\bigr)$ [closed]

To see this question in context click here. How do I deduce the following: \begin{eqnarray} \sum^{k}_{i=1} \dim(V_i) > n(k-1) \implies n>\sum_{i=1}^k\bigl(n-\dim V_i\bigr) \end{eqnarray}