1
vote
1answer
52 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
1
vote
1answer
57 views

Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
8
votes
2answers
617 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 ...
15
votes
5answers
266 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
65 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
45 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
237 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
56 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
162 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
118 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 ...
6
votes
3answers
3k 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 ...
12
votes
3answers
363 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
4k 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 ...