Tagged Questions
0
votes
0answers
31 views
Find two linearly independent functions with a zero Wronskian but a nonzero product.
Known to me examples of L.I. functions having Wronskian=0 have also product=0. One such example was manufactured by Peano about 1890. By the way: Analytic functions with a zero Wronskian are linearly ...
11
votes
7answers
523 views
Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]
This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
12
votes
4answers
400 views
Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis
I am studying for an exam and I have been studying my butt off
during the winter break for it. During the course of my study I have written
down quite a number of tricks, which in my opinion were ...
3
votes
1answer
103 views
Nice applications of the Haar measure
The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is ...
1
vote
0answers
56 views
Partition of open sets in $\mathbb{R}^d$.
Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets.
In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
2
votes
1answer
68 views
Inequalities involving some common functions
I often see the following inequality is used over and over again
$$
1−x⩽e^{−x}
$$
for $x \in \mathbb{R}$, for proving or deriving various statements.
As a layman, I haven't seen this inequality ...
20
votes
1answer
566 views
Expository articles on Analysis and Probability theory
When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
-3
votes
1answer
143 views
Examples of interesting sequences [closed]
Any good series or sequences you have?
For example if you sum the reciprocal of primes this diverges.
As Q is de-numerable, almost any Cauchy sequence we pick will not converge in Q.
Stuff like ...
13
votes
2answers
389 views
open conjectures in real analysis targeting real valued functions of a single real variable
I am hoping that this question (if in acceptable form) be community wiki.
Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may ...
-1
votes
1answer
327 views
Self-Contained Treatments of Stokes's Theorem for Manifolds [closed]
I am seeking to compile a list of textbooks that provide self-contained treatments of Stokes's Theorem in the language of differential forms and manifolds. By "self-contained", I mean the statements ...
8
votes
2answers
628 views
Proofs of the Cauchy-Schwarz Inequality?
How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
20
votes
15answers
2k views
Useful examples of pathological functions
What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples?
For instance, if $f'(a) = b$, then $f(a)$ ...
