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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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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10 views

Finiteness of Assouad-Nagata dimension in neighbourhood

Given a metric space $Y \subset X$ with finite Assouad-Nagata dimension. When is it possible to find a $\epsilon$-neighbourhood (meaning $N_\epsilon = \{x \in X : d(x,Y) \leq \epsilon\})$ which also ...
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22 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
14 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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27 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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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
27 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
37 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
38 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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30 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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1answer
31 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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76 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 ...
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0answers
19 views

Small inductive topological dimension of the real line

I would like to prove that $\mathbb R$ has small inductive dimension $ind \mathbb R = 1$. I think that this statement is equivalent to prove that there is a basis $W$ such that $ind (\partial A) ...
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1answer
37 views

Inductive topological dimension and cross product: for every $A$, we have $ind(A \times B) = ind(A)$ if $ind(B)=0$

Let us denote by $ind X$ the small inductive topological dimension of $X$. How shall I prove that, for every $A$, we have $$ind(A \times B) = ind(A)$$ if $ind(B)=0$?
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1answer
22 views

Equivalence of definitions of dimension in integral domains.

If $R$ is a finitely generated algebra over a field $k$ that is an integral domain, it is known that the Krull dimension of $R$ is equal to the transcendence degree of the field of fractions of $R$ ...
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24 views

Can more 2 dimensional shapes be tessellated in higher dimensions?

I know only triangles, parallelograms, and hexagons can be tessellated on a plane, but could something like a regular pentagon be tessellated infinitely in 3 dimensions or more? I think a bit of a ...
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30 views

How to prove that the dimension of the twisted cubic is 12?

The twisted cubic $C=AX$, $A$ is $4*4$ non-singular matrix. $X=[1\quad t \quad t^2 \quad t^3]^T$. How to prove that the dimension of $C$ is $12$? As I have known that the dimension of $A$ is $15$ ...
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26 views

On the covering dimension of an image under a continuous function

I'm trying to solve the following exercise: Let $X$ be a compact Hausdorff space and let $U_1,...,U_n$ be a cover of $X$ of order $m$. Let $z_1,...,z_n\in\mathbb{R}^N$ for some $N$ be in general ...
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1answer
43 views

If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?

Suppose you have a pair of integral varieties $X$ and $Y$ such that $\dim(X)=\dim(Y)>0$, and there exists a surjective morphism $X\to Y$ between them. I was wondering, if $X$ is affine, does this ...
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20 views

Sufficient condition for integer Hausdorff dimension.

It is pretty much in the title: is there a non-trivial sufficient condition on geometrical shapes that forces the Hausdorff dimension to be an integer ? Most fractals look "complicated" in some way, ...
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18 views

Hyperplane Equation

Would it be correct to say that the function $$ A\cdot T_1+B\cdot T_2+C\cdot T_3=D\cdot x_1+E\cdot x_2+F\cdot x_3+G $$ Generates a hyperplane in 6D? (A through G are constant parameters) Thank you.
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44 views

What is the name of the measurement along a 4th dimensional axis?

Given that measurement along the X, Y and Z axes correspond to the terms "width", "height", and "depth", is there an accepted term for spatial measurement along the W axis when dealing in four ...
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70 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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58 views

Do mathematicians visualise what 4D would look like when they are working with abstract 4D concepts?

I was wondering if mathematicians do truly have a sense of visualising what the fourth dimension would look like in general, such as the fourth dimension as found in hypercubes. I know that we, as 3D ...
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20 views

What is the dimension of the space $W_2$ if $\text{dim}(W_1+W_2) = 2$, $\text{dim}(W_1 \cap W_2) = 1$ and $W_1 = \langle a_1, a_2 \rangle$?

What is the dimension of the space $W_2$ if $\text{dim}(W_1+W_2) = 2$, $\text{dim}(W_1 \cap W_2) = 1$ and $W_1 = \langle a_1, a_2 \rangle$, where $a_1 = (1,0,0)$, $a_2=(2,0,0)$? Okay so I know and ...
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25 views

Can we estimate the dimension of some function space?

Can we estimate the dimension of some function space? e.g. if $X$ is a locally compact Hausdorff space, what is the (linear) dimension of $C(X)$(as a linear space), which represents the set of all ...
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14 views

Cohomological dimension of an orbit space in Alexander-Spanier theory

Take a ring $R$ and define $cd_R(X)$ ("cohomological dimension over $R$") to be the maximum of the integers $m$ such that $H^m(X; R)\neq 0$, where $H^*$ denotes the Alexander-Spanier cohomology. Now ...
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35 views

How can the Hausdorff measure be nonzero?

We have dim$F := \inf \left\{s > 0 : \mathcal{H}^s (F) = 0\right\}$. My question is, with dim$F$ defined as the value where the Hausdorff measure equals zero, then how can ...
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Proof of an inequality about set in four-dimensional space

It is needed to prove that for set $A \subset N^4$ $3\log|A|\le \log|\pi_{1,2,3}(A)|+\log|\pi_{1,2,4}(A)|+\log|\pi_{1,3,4}(A)|+\log|\pi_{2,3,4}(A)|$ where $\pi_{i,j,k}(A)$ is a projection of $A$ on ...
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1answer
55 views

an example of when Hausdorff and box-counting dimensions are equal?

I am new to fractals and dimension theory, so please excuse any errors in my understanding. For a set $F$, let $dim_b (F)$ be the box counting dimension of $F$, and $dim_H (F)$ be the Hausdorff ...
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Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
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1answer
47 views

Proof of fractal dimension of Thomae's function

Thomae's function is defined to be $0$ if x is irrational. Its defined to be $1 \over q$ where $x={p \over q}$ in lowest terms and $q \gt 0$. Its measure is $0$ since the set of rational numbers is ...
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1answer
37 views

Dimension Theorem Corollary

Let $V$ and $W$ be vector spaces with $\dim V = \dim W$. If $T : V → W$ is linear then $T$ is one-to-one if and only if $T$ is onto. But this is true only when the dimensions of $V$ and $W$ are ...
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How does one estimate the Hausdorff measure for arbitrary fractals, and does the constant c in $N=c\epsilon^d$ provide a good estimate?

Background: When one finds the fractal dimension of a fractal in real life, they will generally use the relation $N=c\epsilon^d$ to do so. However, the constant c is almost always neglected in ...
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105 views

Dimension of local rings on scheme of finite type over a field.

In chapter III Hartshorne seems to be using without proof or mention a theorem on the dimensions of local rings of schemes of finite type over a field. I know that for an integral scheme of finite ...
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31 views

Finding dimension of vector spaces

Let V be a vector space and W a subspace of V . Let q : V → V/W be defined by q(v) = v + W for v ∈ V. a) Prove that q : V → V/W is a linear transformation which is onto and show that N(q) = W. b) ...
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Higher dimension leads to different optimization?

It appears z= f(x,y) has a global max/min at another particular (x,y). Using only one independent variable $x$ at fixed $y$ i.e., for z = f(x) I get another max/min point for $x$ optimum point ...
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1answer
48 views

Isn't three-dimensional really three-directional?

This occured to me while writing a mathematics/physics library. Given a 3d shape such as a cube, since it provides a simple demonstration, you have three commonly used measures: length, width, and ...
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1answer
30 views

Dimension of a LOTS in $\mathbb{R}^2$

Equip $\mathbb{R}^2$ with the lexicographic order $(x_1,y_1) < (x_2,y_2)$ iff $x_1 < x_2$ or $x_1 = x_2 \Rightarrow y_1 < y_2$. Consider the induced order topology generated by the subbase ...
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1answer
92 views

Example of affine variety with infinite dimension

It is a typical question in Algebraic Geometry to find a Noetherian topological space with infinite dimension: we can take $X=[0,1]$ and check that. However, it is not so obvious for me to: Find ...
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Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that ...
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Length of a vector in 3 dimension spaces

while I'm studying principal-component analysis, I have misunderstanding about principal component, I was think that it's orthogonal and has a single point,which I mean by single points is if we have ...
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2answers
58 views

Explicit formula for IFS fractal dimesnion

Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or ...
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1answer
55 views

Integral over Fractals with respect to fractal dimension

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the ...
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1answer
57 views

Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
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40 views

A question on Lebesgue's covering dimension

Roughly, a compact, Hausdorff space $X$ has covering dimension $\leqslant n$ if each finite cover $\mathcal{U}$ of $X$ can be refined by a cover $\mathcal{V}$ such that each point $x\in K$ belongs to ...
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89 views

Wheel Graphs and Dimension of Embeddings

I'd like to preface this by saying this is the tip of the iceberg for an optional question for a summer REU program application, so if you think asking this question is in bad taste, let me know and I ...
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prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
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Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
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Compute the dimesion of these subsets of $\mathbb{R}^4$

I need to compute the codimension of the subsets $S_1,S_2,S_3\subseteq\mathbb{R}^4$ given by \begin{align} S_1&=\{(a,b,c,d)\in\mathbb{R}^4:ad-bc\neq0,\},\\ ...