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Questions tagged [big-list]

Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling, and please do not use this as the only tag for a question.

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*easy* examples of fact in one area of maths proven by a different area of maths

Example (something I raised my hand to ask when I was in secondary school): How do you know that $\dfrac{a!}{b!(a-b)!}$ is necessarily an integer whenever $a$ and $b$ are natural numbers such that $a&...
Chris Sanders's user avatar
1 vote
1 answer
60 views

Examples of Complex Manifolds

I'm trying to learn about Complex Manifolds, and something that has proven a stumbling block is the paltry number of examples provided. The best of the book I found was Huybrechts Complex Geometry ...
Derivative's user avatar
  • 1,568
16 votes
3 answers
580 views

Probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation.

My question is: What are some examples of probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation? Context Some probability questions have answer $\frac{1}{2}$, and - as ...
Dan's user avatar
  • 22.9k
10 votes
4 answers
556 views

List of geometric theorems linked by two squares

I'm trying to create a classification for geometric theorems that relate to two squares As a type of organization and classification And the curiosity to explore I have collected some theorems of this ...
زكريا حسناوي's user avatar
4 votes
0 answers
99 views

Infinite products that are equal to their geometric product integral

Background In the following question and references therein, a number of "Sum equals integral" identities are described. For instance, we have $$ \sum_{n = -\infty}^{+\infty} {\rm sinc} (x)^{...
Max Muller's user avatar
  • 7,058
5 votes
2 answers
146 views

Organizing the content of Euclidean geometry with pictorial mind maps

There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to ...
زكريا حسناوي's user avatar
13 votes
5 answers
522 views

Statements which feel like they shouldn't be first-order expressible, but are

I recently had the opportunity to study some (very basic) model-theory, and that made some theorems in ring theory become immediately more interesting. For example, one characterization of the ...
delta_phi's user avatar
  • 227
10 votes
2 answers
320 views

Applications of Linear Programming to pure mathematics

This semester I'm taking a course in Linear Programming. While the topic is very interesting, all the applications I can find about this topic seem to be outside of mathematics. What are some ...
Carla_'s user avatar
  • 149
1 vote
0 answers
84 views

Searching for textbooks that teach math differently from other textbooks about the same subject. [closed]

Famous examples would be LADR (which to an extent eschews determinants) and Aluffi's Algebra (which involves some category theory). I would like more books in this vein, that either teach differently ...
valley's user avatar
  • 19
1 vote
0 answers
58 views

Good books/resources for learning how to visualize Mobius transformations

Posts like this one have always baffled me. I have taken a few complex variables courses in my time and it seems that every time I take a course like this, either the textbook isn't great at ...
Grigor Hakobyan's user avatar
3 votes
0 answers
125 views

Fermat's (hypothetical) erroneous proof

Until Wiles' proof of Fermat's last theorem all proposed proofs have been erroneous. It is not known which proof Fermat himself had in mind - but it is assumed that it was erroneous, too. Have there ...
Hans-Peter Stricker's user avatar
3 votes
2 answers
299 views

Overview of basic results in Stochastic Calculus

Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus? These can be in the form of resources (preferably accessible online) as well as directly writing out ...
FD_bfa's user avatar
  • 4,031
3 votes
2 answers
79 views

Examples of natural proofs by induction exercises which needs 3 or more base cases?

I am teaching proof by induction next week, so I am looking for new some good examples. I'd like some natural (so not just some linear recursion depending on three previous terms) problems where three ...
Per Alexandersson's user avatar
4 votes
2 answers
91 views

Concrete examples of internal categories (other than small categories)

I understand the definition of an internal category, but so far I haven't found examples of internal categories (other than categories internal to $\mathrm{Set}$, which are just the small categories) ...
Anakhand's user avatar
  • 2,592
1 vote
4 answers
88 views

What methods can be used to solve $y'=xy;y(0)=2$?

This is a very basic separable differential equation and can be easily solved as such: $$\frac{dy}{dx}=xy;\,y(0)=2$$ $$\frac{dy}{dx}=xy$$ $$\frac{dy}{y}=xdx$$ $$\int\frac{dy}{y}=\int xdx$$ $$\ln|y|=\...
Dylan Levine's user avatar
  • 1,596
0 votes
1 answer
92 views

List of open conjectures on fractals [closed]

I want a list of open conjectures on fractals. Preferably easy to understand and low dimensions. Before you say Eremenko's conjecture ( https://en.wikipedia.org/wiki/Escaping_set ): https://arxiv.org/...
mick's user avatar
  • 16k
3 votes
2 answers
256 views

What are fun mathematical facts for non-mathematicians? [duplicate]

I like to spend my life with mathematics. I think it is the best thing I can do in my life. However, I have great difficulty explaining what I am doing to non-mathematicians, even educated ones. For ...
boyler's user avatar
  • 335
1 vote
2 answers
113 views

homomorphisms from a (semi)local ring to a ring with infinitely many maximal ideals

In continuation of the this question: homomorphism from a (semi)local ring to $\mathbb Z$. I tried to construct (unital) homomorphisms from a (semi)local ring to a ring with infinitely many maximal ...
user1401's user avatar
2 votes
0 answers
51 views

The use of threads and nails in geometry [closed]

For a long time, I have been thinking about starting to study geometry that uses strings and nails, but I was postponing the idea. In fact, strings and nails are more advanced tools than a ruler and a ...
زكريا حسناوي's user avatar
0 votes
3 answers
4k views

What are the interesting ways to write $2024$ and some facts about $2024$? [closed]

What are the interesting ways to write $2024$ and some facts about $2024$? We know that $2024$ is coming in $48$ hours, so I decided to ask this. It is well known fact that $45^2=2025$, so $2024$ can ...
Thirdy Yabata's user avatar
1 vote
1 answer
107 views

Listing methods to prove that two rings are not isomorphic [closed]

Really, this question comes down to listing propeties that are preserved by ring isomorphisms. Off the top of my head, I can think of: cardinality of the ring commutativity the order of elements ...
Adil Raza's user avatar
12 votes
2 answers
397 views

$q$-analog of Number Theory [closed]

The main motivation behind this is to see whether the 'magic' of q-analogs can be felt in number theory. Obviously for q-analogs to be applied to number theory the parametrization in $q$ must yield a ...
Mako's user avatar
  • 546
1 vote
1 answer
246 views

Ways to "milk" the integral $\int^{1}_{0}\ln\left(\phi^{x}-\phi^{-x}\right)\ln\left(\phi\right)dx=-\frac{\pi^2}{20}$

After reading the MSE post on "Integral Milking", my first instinct was to try it out on one of my favorite integrals: $$\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$$...
Dylan Levine's user avatar
  • 1,596
3 votes
1 answer
82 views

Rigorizing the notion of fundamental objects for topological properties

I've noticed many topological properties have a fundamental object for which we have to put in some elbow grease to establish it from scratch, after which there is a slew of theorems allowing us to ...
Display name's user avatar
  • 5,154
0 votes
0 answers
30 views

Open problems in Lie symmetry theory

What are some famous open problems and conjectures in Lie symmetry theory? Is there some kind of list of these problems available or possibly a historical survey of the development of theory of ...
wurd's user avatar
  • 1
2 votes
1 answer
230 views

Let $S\subseteq G$ for a group $G$. What is the relationship between $\langle S\rangle_G$ and ${\rm ncl}_G(S)$?

This is a big-list question that I suppose could be a community wiki post. If this is too broad or otherwise a bad question, I'm sorry. The Question: Let $S\subseteq G$ for a group $G$. What is the ...
Shaun's user avatar
  • 45.2k
10 votes
6 answers
1k views

Examples of small categories

Most categories we encounter in daily mathematics life ($\mathrm{Grp}$, $\mathrm{Ab}$, $\mathrm{Ring}$, $\mathrm{Top}$, $\mathrm{Set}$...) are not small. I've been looking around here and elsewhere on ...
Anakhand's user avatar
  • 2,592
1 vote
1 answer
151 views

Elementary examples of Yoneda embeddings

Suppose you trying to sell the idea of the Yoneda embedding to perhaps a rather mixed bunch of students (so you can't presuppose too much mathematical background). Still you can say Think of a group-...
Peter Smith's user avatar
  • 54.8k
1 vote
0 answers
35 views

Intersection Of Common Notation [closed]

Just for fun, I'm wondering if anyone can give examples of common symbols/variables that are used in completely different ways depending on the context. Some examples I can think of off the top of my ...
EDS's user avatar
  • 592
2 votes
1 answer
133 views

Odd-sounding "everyday" applications of the Handshaking Lemma

First of all, a convention. Let $(V,E)$ be a graph with degree function $\operatorname{deg}:V\to\mathbb{N}_0$. There are three results which get referred to as the Handshaking Lemma $\sum_{v\in V}\...
JP McCarthy's user avatar
  • 8,419
0 votes
0 answers
46 views

What are applications of changing limit and differentiation/integration?

I know the following theorems but don’t know their usefulness. If a series $\{f_n\}$ of Riemann integrable functions on $[a, b]$ uniformly converges to $f$, $f$ is Riemann integrable and $\lim\limits_{...
MathMan's user avatar
  • 109
3 votes
1 answer
222 views

Times when you want $(\cdot)^2$ and not $|\cdot|^2$ when porting from $\mathbb{R}$ to $\mathbb{C}$

Over $\mathbb{R}$, the square, $(\cdot)^2$, and usual square-norm, $|\cdot|^2$, functions agree for all inputs and are occasionally used interchangeably. For instance, when computing the variance of ...
Integrand's user avatar
  • 8,467
37 votes
9 answers
3k views

Looking for (overkill) usages of indicator functions

I am going to give a presentation about the indicator functions, and I am looking for some interesting examples to include. The examples can be even an overkill solution since I am mainly interested ...
MR_BD's user avatar
  • 5,952
0 votes
0 answers
52 views

Find an example where there is a variable that affects the result, unlike expected

I don't know if this site is a suitable place to look for examples, but I have a lot of ideas that I would like to find a good example of, and I try to organize these things in a later book. I hope I ...
زكريا حسناوي's user avatar
1 vote
0 answers
196 views

What are all the tetration extension methods?

Tetration is the next step in our regular operations. Addition, multiplication, exponentiation, tetration. It is constructed by repetitive exponentiations. "$a$ tetration $b$" is written $^{...
Pierre Carlier's user avatar
78 votes
5 answers
8k views

Theorems that disappointed mathematicians

I was looking for a list of theorems that disappointed mathematicians, in the sense that some disgruntlement regarding the fact that the theorem in consideration being true/false was notable. An ...
QED's user avatar
  • 877
2 votes
2 answers
193 views

Research monographs and open problems in universal algebra

I am someone who is very interested in the mathematical subfield of universal algebra. I want to know, what are some significant open problems in universal algebra? I would like a list of such ...
user107952's user avatar
  • 20.7k
0 votes
2 answers
149 views

Theorems of elementary geometry that can be generalized to inner product space over $\mathbb{R}$ [closed]

In inner product spaces on $\mathbb{R}$, length and angles can be defined. Hence, something similar to elementary geometry can be done there. For example, the parallelogram law can be generalized to ...
Htmm's user avatar
  • 1
-1 votes
1 answer
57 views

List of Cartesian coordinates laws

When I search online for any list of laws of Cartesian coordinate geometry, I find that it includes quite a few laws such as the law of distance between two points, the law of distance between a point ...
زكريا حسناوي's user avatar
0 votes
0 answers
54 views

A large reference or list of examples of Euclidean geometry and imaginary states

Mathematics has always been concerned with extending existing concepts to more comprehensive cases, I am a big fan of what I would call "imaginary geometry" might be a bad name, I don't know ...
زكريا حسناوي's user avatar
1 vote
0 answers
153 views

Are there any inquiry based Calculus books?

This question may have been asked a bunch of times, but I have not found exactly what I was looking for. I say calculus, but mathematical/real analysis works as well as long as it does not have ...
Not Meta's user avatar
62 votes
27 answers
7k views

What are some conjectures of your own?

Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow), which gather a lot of attention to the site are about ...
2 votes
0 answers
61 views

Iff propositions where both directions require choice?

Recently, I have been revising a basic course on noncommutative rings and modules over them. One proposition proven early on is all left modules over $R$ are free iff $R$ is a division ring and an ...
Isky Mathews's user avatar
  • 3,235
3 votes
0 answers
59 views

Examples of existence proof by comparison of cardinalities

There is a famous proof of the existence of a transcendental number by comparison of cardinalities. I would like to know more examples of existence proof by comparison of cardinalities.
BonBon's user avatar
  • 397
3 votes
1 answer
151 views

Reference Request: Open-Access Linear, Non-Linear, etc Optimization Textbooks

Currently, I am building a book report/reviews in an effort to convince some professors to change their in-class textbooks to free open-access books to reduce the burden of cost unto the students ...
Miss Mae's user avatar
  • 1,641
2 votes
0 answers
54 views

How to get the number of path components of the set of roots of a polynomial

Sorry in advance for the open ended question. Let $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ be a polynomial. What techniques are there for obtaining the number of path components of $f^{-1}(0)$ from ...
Amr's user avatar
  • 20.1k
2 votes
0 answers
60 views

Use cases for $L^p$ and $l^p$ spaces where $p\neq 1,2,\infty$

Soft question: $L^1,L^2,$ and $L^\infty$ spaces all have many practical uses and an easy intuition behind them (Along with the $l^1$, etc. versions). Just for visualization, I was playing around ...
Alan's user avatar
  • 16.6k
0 votes
1 answer
93 views

names for $f^{-1}(0)$

What are commonly-used names for $f^{-1}(0)$, where $f: X \to Y$ for $Y$ some algebraic structure with $0$? I am particularly thinking of the case where $Y = \mathbb{R}$, where I was expecting that ...
Jacob Maibach's user avatar
1 vote
0 answers
94 views

Integrals with massive amount of cancellation (positive and negative portions of integrand cancel out almost perfectly)

In Gamelin's Complex Analysis, there are exercises/examples (pg. 201-202) of the form $$\int_{-\infty}^\infty \frac{P(x)}{Q(x)} \cos(ax) dx$$ The first "trick" is to use complex analysis: ...
D.R.'s user avatar
  • 8,711
16 votes
5 answers
1k views

What are examples of Halmos's claim that a single small concrete special case can capture every instance of a concept of great generality?

Paul Halmos states: It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case. What are examples of ...
SRobertJames's user avatar
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