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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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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4answers
1k views

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 ...
29
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1answer
945 views

Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class ...
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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 ...
17
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5answers
362 views

Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples

In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen ...
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$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$

Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ ...
14
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7answers
807 views

Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

The notion of having a number $a \in \mathbb{R}_{\geq 0} $ associated to any metric space is described by the definition of a "Hausdorff Dimension". I was wondering if work has been done on spaces ...
13
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2answers
297 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 $x$ ...
11
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1answer
375 views

Fractional versions of euclidean space?

This is going to be a somewhat vague question, but I'll be happy if you indulge me. Euclidean space $\mathbb{R}^n$ is equipped with a lot of nice (algebraic, metric, topological,...) structure and ...
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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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Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
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1answer
208 views

Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
9
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2answers
675 views

Why should Gaussian noise have fractal dimension of 1.5?

In a paper I'm trying to understand, the following time series is generated as "simulated data": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,\ldots,N)$$ where $Z(j)$ is a Gaussian noise ...
8
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239 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 ...
8
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1answer
86 views

Continuous one-to-one mapping from a subset $K \subset \mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f\colon \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a continuous function and $K\subset \mathbb{R}^n$ a subset of positive Lebesgue measure. Is it possible that $f$ is one-to-one on $K$? If $K$ ...
8
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1answer
745 views

For an $n$-dimensional object, how many types of holes are possible?

Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
8
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1answer
152 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 ...
7
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2answers
663 views

Is zero the center of the numeric sequence?

The numeric sequence has symmetry on zero, with equal infinities of cancelling out + and - values on either side. Can numbers be said to have different centers of symmetry than zero? Is it possible ...
7
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1answer
187 views

Vector spaces with fractional dimension

Can the notion of vector space or algebra over a field be meaningfully extended to fractional dimensions, so that for example $\mathbb{R}^{-2/3}$ makes sense? Has this been explored somewhere? I know ...
7
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1answer
208 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 ...
6
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1answer
306 views

Fractals - when the number of seed shapes that can fit into the scaled-up copy is non-integer.

I've heard people say (for eg. here) that the dimension of fractal patterns (particularly, in this question, Lindenmayer fractals) can be formulated as follows: $$D=\frac{\ln N}{\ln S}$$ Where $N$ ...
6
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1answer
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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 ...
6
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1answer
226 views

Self-similar fractal dimension of unsymmetrial fractal

As far as I know, the following fractal has a self-similar fractal dimension of $D = -\log(3) / \log(1/2) = 1.5850$ But what is the fractal dimension of the following fractal (4 times the fractal ...
6
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1answer
248 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 ...
6
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2answers
692 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 ...
6
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112 views

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 ...
5
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3answers
2k 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 ...
5
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2answers
78 views

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 ...
5
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166 views

Does interval spacing effect Hausdorff dimension of Cantor set?

Let $C=\bigcap_{j=0}^{2^n}C_j$, $C_0=[0,1]$, and the intervals in the construction of each stage of $C_j$ consists of removing the center 1/3 from the $j-1$ stage intervals. In other words, the ...
5
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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 ...
5
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1answer
249 views

Dimension of subspace is greater then dimension of space

If $X$ is a topological space then it's (covering) dimension is defined as a minimal number $n$ such that for every finite open cover $\{U\}$ of $X$ there is a finite open cover $\{V\}$ of $X$ that ...
5
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2answers
227 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 ...
5
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135 views

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 ...
5
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1answer
86 views

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 ...
5
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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 ...
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93 views

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 ...
4
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2answers
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Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
4
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1answer
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!
4
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115 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 ...
4
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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 equivalence does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
4
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3answers
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Confusion related to curse of dimensionality in k nearest neighbor

I have this confusion related to curse of dimensionality in k nearest neighbor search. It says that as the number of dimensions are higher I need to cover more space to get the same number of ...
4
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3answers
154 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 ...
4
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1answer
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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
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1answer
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algebraic geometry exercise: infinite subset is dense

A hypersurface $C \subset \mathbb{A}^2$ we call a plane curve. Show that any infinite subset of an irreducible plane curve $C \subset \mathbb{A}^2$ is dense in $C.$ Note. We call hypersurface the ...
4
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1answer
295 views

Doubts about Hausdorff measure.

I am studying real analysis and I have some problems in understanding properties of Hausdorff measure. Let $\mathcal{E}_\delta$ be collection of subsets of $\mathbb{R}^N$ whose diameter is less then ...
4
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1answer
157 views

If the special fiber of a flat morphism is reduced, then any other fiber is reduced?

Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...
4
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1answer
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 $m$ ...
4
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2answers
115 views

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 ...
4
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1answer
63 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 / ...
4
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2answers
123 views

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 ...