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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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 ...
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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 ...
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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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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 ...
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75 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 ...
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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 ...
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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 ...
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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 equivalance does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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107 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 ...
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185 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 ...
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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 ...
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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 ...
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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 ...
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173 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 ...
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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 ...
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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 ...
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683 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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? ...
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$T:V\rightarrow W$ 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 ...
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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?
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Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
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$\sum^{k}_{i=1} \dim(V_i) > n(k-1) \implies n>\sum_{i=1}^k\bigl(n-\dim V_i\bigr)$ [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}
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Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
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Dimensionality and Subspace Existence: A Potential Outlet for Disquisition

The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?).... Initially, my intuition said the ...
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1answer
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Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$

SEE AUTHOR'S ANSWER BELOW So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
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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 = ...
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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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Poles of formal power series (Hilbert-Poincaré series)

How are poles and orders of poles of formal power series defined? The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, ...
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423 views

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 ...
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Measuring Vitali sets

If $V$ is a vector space and $m$ is the counting measure, then $$\dim(V) = \inf \{m(U) : U \subset V, \text{ span}(U) = V\}.$$ Given a measure space $(V, \mathcal M, \mu)$ such that $V$ is a vector ...
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Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to ...
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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 ...
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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 ...
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How to apply other than trivial dimensions?

I only used natural dimensions so far but I understand there could be Negative dimension with application e.g. dimension -2 definition and usage Non-integer dimension with application e.g. dimension ...
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Matrices to model 3D object

I'm toying around with an algorithm to determine placement of 3D objects into a larger 3D space. I immediately thought of using matrices. It's been some years since my Linear Algebra courses. I was ...