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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Hausdorff dimension of graph of function

This question came up on an exam Decide the Hausdorff dimension of the graph of the following function for $x>0$ $$y = \log(1+x)\sin\frac{1}{x}$$ In the course, we only touched upon the subject ...
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169 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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30 views

Does finite covering dimension imply local compactness?

I have a space which is not locally compact and I'm trying to see if I can say anything about the dimension of the space. I suspect that it is not finite dimensional but I have thus far been unable to ...
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Where do extra dimensions in gradient come from?

The gradient of a scalar function $f\colon \mathbb{R}^n \to \mathbb{R}$ is a vector-valued function $\nabla f\colon \mathbb{R}^n \to \mathbb{R}^n$. Since applying a function can't increase ...
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Intersection of 2 continua with an arbitrary number of dimensions using matrices

If I have 2 continua (i.e. a line, plane, space, hyperspace etc.) of an arbitrary number of dimensions, each described parametrically as: x = xs + x1*t + x2*u + x3*v + ... + xn*k y = ys + y1*t + ...
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Is it possible to construct a smooth curve with fractional Hausdorff dimension?

It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true? If a curve has a fractional Hausdorff ...
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1answer
11 views

rectangular paddock, dimensions, maximise area it encloses

Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot, ...
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125 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
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271 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 ...
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27 views

How many minimally sides are needed to fully enclose a volume in an $n$-dimensional spaece?

Imagine I have some unbendable*, but cuttable metal-plates. One side is silver, the other one is blue. Now I want to use them to build somekind of case (with no opening) that is fully blue. It just ...
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15 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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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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154 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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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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24 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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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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39 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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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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39 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
101 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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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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39 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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79 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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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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97 views

Can an n dimensional object cover an n+1 dimensional object?

Is it possible for an $n$-dimensional object to ever cover an $(n+1)$-dimensional object? For example, could a square ever cover a cube? Note: Definition of "cover" here means to completely cover the ...
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25 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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27 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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45 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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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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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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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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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 ...
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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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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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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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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 $x$ ...
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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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16 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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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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59 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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56 views

Intuitive meaning of fractal dimension.

I'm studying M. Barnsley's book 'Fractals Everywhere', but I'm stuck in the chapter 'Fractal Dimension'. Suppose $(X, d)$ is a complete metric space and let $A \in \mathcal{H}(X)$ be a nonempty ...
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1answer
51 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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65 views

The existence of a measure of finite energy implies a lower bound on Hausdorff dimension

What is the significance of $\mu(x)=0$ and the use of continuity this proof? I am not quite sure about the general direction in the second paragraph.
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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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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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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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35 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) ...