Tagged Questions
2
votes
1answer
63 views
What is the intuition between 1-cocycles (group cohomology)?
This is, I'm sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles?
Let me explain a bit. Given an action of a group $G$ on another ...
14
votes
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 ...
4
votes
0answers
51 views
Quotient-lifting properties
I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies
Let $N\triangleleft G$. Then ...
1
vote
0answers
42 views
Duality between $[G,G]$ and $Z(G)$? [duplicate]
Possible Duplicate:
Center-commutator duality
Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator \begin{equation}
...
8
votes
2answers
148 views
What was Klein working on when he “replaces his Riemann surface by a metallic surface”?
I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting:
Look at Professor Klein: he is studying one of the most abstract questions of ...
2
votes
1answer
44 views
So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?
For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation}
H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\},
\end{equation} where $\mathcal{S}'$ is the space of ...
4
votes
2answers
113 views
The Duality Functor in Linear Algebra
I'm trying to gain an intuitive understanding of the following construction:
For any vector space $M$ over a field $R$, one can define the algebraic dual of $M$ as $M^* := \mathsf{Hom}(M, R)$ and ...
3
votes
3answers
98 views
Intuition for Coconstant morphisms
A constant morphism $f \in \mathrm{Hom}(X,Y)$ is a morphism such that for any object $Z$ and any morphisms $g,h \in \mathrm{Hom}(Z,X)$, $f \circ g = f \circ h$. This is very easy to grasp and one can ...
5
votes
2answers
1k views
Examples for proof of geometric vs. algebraic multiplicity
Here you see a supposedly easy proof of a well-known theorem in linear algebra:
Although I know I should understand this, I don't :-(
Obviously there are too many indices and stuff, so I don't see ...
11
votes
3answers
272 views
Nasty examples for different classes of functions
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Usually when proving a theorem where $f$ is assumed to be continuous, differentiable, $C^1$ or smooth, it is enough to draw intuition by assuming ...
9
votes
6answers
2k views
How are eigenvectors/eigenvalues and differential equations connected?
In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts.
I learned that ...
