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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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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3answers
50 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
78 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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0answers
11 views
Cubic interpolation in arbitrary dimension?
Consider a $N$-dimensional space discretized with a regular cubic grid of $n^N$ cubes, each cube containing the value of a function $f$ in its center. How to correctly interpolate $f(x, y, z)$ using ...
2
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0answers
38 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
67 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
43 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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2answers
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$T:V\rightarrow W$ such that $R(T) \subset W'$ is a subspace of ${\cal{L}}(V,W)$ [closed]
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ and $W'\subset W$ a subspace, then the subset ${\cal{L}}(V,W)$ consisting of all linear maps $T:V\rightarrow W$ such that $R(T) \subset W'$ ...
4
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4answers
95 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
77 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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1answer
30 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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1answer
25 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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2answers
37 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?
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4answers
100 views
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
48 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 ...
2
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2answers
57 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 = ...
21
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0answers
334 views
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 nontrivial ...
2
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1answer
48 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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1answer
20 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 ...
3
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2answers
113 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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0answers
51 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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1answer
67 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 ...
4
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1answer
51 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 ...
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0answers
29 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 ...
3
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1answer
117 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
62 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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0answers
43 views
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
53 views
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
37 views
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
204 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 ...
2
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1answer
59 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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0answers
98 views
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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1answer
78 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 ...
2
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1answer
63 views
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
106 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?
2
votes
1answer
127 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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2answers
118 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 ...
3
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1answer
42 views
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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2answers
44 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
73 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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0answers
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Density of a multifractal distribution
I am trying to grasp the concept of density of a multifractal. So I start from the simple case of a line. Let's assume I have a uniform distribution of points on a line and I center a cubic box in the ...
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1answer
25 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 ...
2
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1answer
139 views
Linear Algebra Question ( rank of matrix )
Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively
prove $\operatorname{rank}(\mathbf{PA}) = ...
3
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1answer
74 views
Prime ideals of height less than the dimension
Let $A$ be a noetherian local ring with maximal ideal $\mathfrak{m}$ of height $d$, and suppose $\mathfrak{p}_1, \ldots, \mathfrak{p}_s$ are prime ideals of height $i - 1 < d$. It's quite clear ...
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1answer
65 views
Related questions on the Hausdorff dimension and local dimension of a Cantor set
Suppose $I$ is an interval on a Cantor tree with $m$ children $I'$, each of length $\varepsilon$. I have that $\sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow ...
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2answers
229 views
Why is it that I cannot imagine a tesseract?
I try hard to "visualise" (say "imagine") a tesseract but I can't.
Why is it that I can't?
This may be a question for a scholar of some other discipline and not for a mathematician, e.g. ...
3
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1answer
72 views
Cutting a $n$-dimensional cubic cake
Given a cubic cake, defined as $\{(x,y,z)|0\leq x,y,z\leq 1\}$.
We cut it by the planes
$p_1\leftrightarrow x=y$
$p_2\leftrightarrow y=z$
$p_3\leftrightarrow x=z$.
How many pieces will we have ...
7
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1answer
140 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 $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
2
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0answers
123 views
Canonical $\pi$ dimensional space?
Can we talk about a canonical space of dimension $\pi$? Is there anything like $\mathbb R^\pi$?
Have anyone met any fractal of dimension $\pi$?
4
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1answer
115 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 $0$, ...
