All Questions
Tagged with big-list ca.classical-analysis-and-odes
20 questions
99
votes
28
answers
14k
views
Probabilistic proofs of analytic facts
What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
- 91.8k
2
votes
0
answers
158
views
What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
- 4,829
31
votes
13
answers
6k
views
Classic applications of Baire category theorem
I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
- 3,317
64
votes
16
answers
13k
views
How helpful is non-standard analysis?
So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...
- 4,713
63
votes
7
answers
5k
views
What well known results with countability assumptions can be naturally extended to uncountable settings?
In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in ...
CommunityBot
- 1
27
votes
29
answers
30k
views
Alternative undergraduate analysis texts
Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results ...
- 63.9k
5
votes
1
answer
2k
views
Examples and importance of Embedding (and Non-Embedding) Theorems
An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
CommunityBot
- 1
43
votes
6
answers
2k
views
Are there "natural" sequences with "exotic" growth rates? What metatheorems are there guaranteeing "elementary" growth rates?
A thing that consistently surprises me is that many "natural" sequences $f(n)$, even apparently very complicated ones, have growth rates which can be described by elementary functions $g(n)$ ...
- 623
38
votes
26
answers
57k
views
Text for an introductory Real Analysis course.
Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
- 8,842
24
votes
12
answers
3k
views
Constructions unique up to non-unique isomorphism
1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...
- 35.5k
191
votes
34
answers
81k
views
What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
- 10.5k
16
votes
2
answers
539
views
Surprising appearances of Painlevé transcendents
What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations?
I have seen examples from enumeration of classes ...
- 401
8
votes
3
answers
1k
views
Uses of Divergent Series and their summation-values in mathematics?
This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see.
Dear MO-Community,
When I was trying to find closed-form representations for odd zeta-...
- 5,342
52
votes
22
answers
19k
views
Interesting Calculus Questions/Exercises
I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...
- 39.9k
35
votes
19
answers
9k
views
Interesting applications (in pure mathematics) of first-year calculus
What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...
CommunityBot
- 1
8
votes
4
answers
2k
views
NP-hard problems in linear algebra and real analysis [closed]
I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...
- 2,803
15
votes
4
answers
3k
views
Statements which were given as axioms, which later turned out to be false.
I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom....
- 47.3k
4
votes
10
answers
1k
views
Proving theorems by using functions with fixed points.
I am trying to get a better feel for solving questions where creating a function with a unique fixed point is the crux of the proof.
In particular, the Inverse Function Theorem as well as the ...
CommunityBot
- 1
1
vote
2
answers
605
views
How many ways can we characterize gamma function?
First let's state a well-known characterization of gamma function.
If f is a positive function on positive real numbers such that:
(1).f(x+1)=xf(x);
(2).f(1)=1;
(3).logf is convex,
then f(x) is gamma ...
- 5,935
7
votes
3
answers
2k
views
What are some interesting sequences of functions for thinking about types of convergence?
I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
