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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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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308 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 ...
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3answers
101 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 ...
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254 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
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1answer
115 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 ...
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1answer
129 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
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3answers
980 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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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
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1answer
131 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
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1answer
90 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 ...
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66 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 ...
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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 ...
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134 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 ...
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1answer
183 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 $x$ ...
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78 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 ...
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1answer
140 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 ...
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2answers
53 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? ...
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139 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 ...
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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?
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49 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?
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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}
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1answer
71 views

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
77 views

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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2answers
5k views

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 ...
2
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1answer
118 views

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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2answers
536 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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154 views

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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63 views

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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117 views

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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34 views

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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1answer
185 views

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 ...
3
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1answer
176 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 ...
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Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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1answer
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Dimensions of Matrices Range (equalities).

I’d like to find range equalities. Considering the following: $$ A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T \\ $$ I would like to find the function $f$ for each equality above. $$ dim( R(A) ) = f( R(B) , ...
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1answer
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Trying to prove $\text{Hdim}(\bigcup X_i)=\sup_i \text{Hdim}(X_i)$

Suppose $X=\bigcup_i X_i$ is a countable union. I'm trying to prove a statement which wikipedia says follows directly from the definition of Hausdorff Dimension: ...
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5answers
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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 ...
2
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1answer
94 views

Dimension of graphs (Differential Geometry)

I have a rather basic question about the dimension of a graph mapped by the exponential map. The problem is I can't really visualize it. It starts like that: Let $M$ be a smooth manifold of ...
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Hilbert (polynomial) dimension and dimension of a support of a module

$\newcommand{\Supp}{\mathrm{Supp}}$ $\newcommand{\Ann}{\mathrm{Ann}}$ Let $X$ be an affine algebraic variety (over a field $K$, can assume it is algebraicaly closed), $M$ a finitely generated ...
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106 views

Injective endomorphism and Hilbert dimension

Let $\mathcal{O}$ be a finitely-generated $K$-algebra where $K$ is a field and let $M$ be a finitely-generated $\mathcal{O}$-module. For every good filtration $0 = M_0 \subset M_1 \subset M_2 \subset ...
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1answer
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Transcendence degree of $K[X_1,X_2,\ldots,X_n]$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every ...
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1answer
237 views

Hausdorff dimension of a smooth manifold

I read a book about fractal stating that without proof: every $m$-dimension $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove it?
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185 views

Krull dimension of the injective hull of residue field

Let $(R,\mathfrak{m})$ be a noetherian local ring, and $E=E_R(R/\mathfrak{m})$ the injective hull of $R/\mathfrak{m}$. What do we know about the Krull dimension of $E$? Thank you.
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396 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": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$ where $Z(j)$ is a Gaussian noise with ...
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n points and (n-1) dimensions

2 points make a line -1D 3 points at most make a plane -2D similary can we find $n$ points making up a $n-1$-dimensional object in $\mathbb{R}^n$?
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158 views

Proof that the Hausdorff Dimension of Liouville Numbers is zero.

I am studying a proof that the set $L$ of Liouville numbers in $[0,1]$ has Hausdorff dimension zero. $L=\lbrace x\in [0,1]: \forall n\in \mathbb{N}, \exists p,q\in \mathbb{Z}, q>1, \text{and such ...
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1answer
120 views

Planar cross section of leech lattice?

I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?
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26 views

Dimension of coefficents in a density equation

The density throughout a composite material is given by $T(x, y, z) = Axy^2 + Bxz^3 + Cy^2z^3,$ where $x$, $y$ and $z$ are the cartesian coordinates of the position inside the material. (a) Find the ...