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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8
votes
3answers
427 views

Instructive examples of elegant, clear, rigorous, terse, but “non-dull” mathematical prose

On the "About" page of the Mathgen project one can read: "More seriously, I think this project says something about the very small and stylized subset of English used in mathematical writing. ...
73
votes
5answers
2k views

“Advice to young mathematicians”

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the ...
10
votes
11answers
418 views

Get the numbers from (0-30) by using the number $2$ four times

How can I get the numbers from (0-30) by using the number $2$ four times.Use any common mathematical function and the (+,-,*,/,^) I tried to solve this puzzle, but I couldn't solve it completely. Some ...
1
vote
0answers
110 views

What is a Toy Model for the mathematician's practice? Definition and examples

Wikipedia says Toy model (physics): "In physics, a toy model is a simplified set of objects and equations relating them so that they can nevertheless be used to understand a mechanism that is also ...
12
votes
8answers
2k views

What properties are true for “almost all real numbers”? [closed]

Over the years, I remember hearing of a lot of proofs of properties that applied to "almost all real numbers", that is the exceptions to the rules had measure zero. But I can't remember many of them. ...
1
vote
1answer
70 views

Fixed Point Equivalences of Axiom of Choice

Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular ...
23
votes
6answers
517 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
44
votes
14answers
3k views

Largest “leap-to-generality” in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that ...
90
votes
22answers
10k views

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
-1
votes
1answer
64 views

What is an elementary yet important application of matrix in finance?

What is an elementary yet important application of matrix in finance? I have difficulty to read anything intermediate/advanced associated with this topics, hopefully I can find something interesting ...
23
votes
5answers
369 views

Examples of Mathematics in Court

In court trials, natural sciences such as physics and biology routinely make an appearance, e.g. when estimating the speed of a vehicle based on impact damage or trying to deduce from the condition of ...
0
votes
0answers
73 views

Free online mathematical tools to solve step by step problems

I am looking for online mathematical softwares which gives stepwise solution of problems.So far I have been able to find Symbolab but its useless unless the problem is very simple.
3
votes
0answers
101 views

Exercises in Topological K-Theory (Atiyah)

I'm currently working through Michael Atiyah's K-Theory. The main problem I'm finding with it is that it does not have any exercises. Does anyone have a good collection of exercises to go along with ...
2
votes
1answer
70 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
0
votes
0answers
33 views

List of functions $\chi_{s,a}(n)$ defined on a Group such that $\chi_{s,a}(n)\in{s,a}$ and depending on the parity

Question Let $(G,\cdot,e)$ be a non-commutative group and $s,a \in G$ .I'm looking for interesting functions $\chi_{s,a}:\Bbb N \rightarrow G$ witht this property $$\chi_{s,a}(n)= \begin{cases} s, ...
25
votes
13answers
684 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
5
votes
1answer
117 views

Discrete math problems

I am a high school student interested in thinking about math. I don't know a lot of high-powered math (I only know up to calculus), instead I focus on discrete topics related to math Olympiads ...
6
votes
1answer
160 views

Ramanujan's False Claims

"During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these ...
114
votes
22answers
18k views

Examples of mathematical discoveries which were kept as a secret

There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret. For example it is completely expected that if some mathematician find ...
4
votes
6answers
1k views

The Largest Gaps in the History of Mathematics

Edit: Based on the useful comments below. I edited the original post in order to seek for other important historical gaps in mathematics. Mathematics is full of the historical gaps. The first type ...
1
vote
0answers
114 views

Equivalence relation over groups $a\asymp_sb :\rightarrow\exists n\in\Bbb Z:as^n=b$: terminology and decision problem

Let's define this relation over the elements of an infinite group $(G,\cdot,e)$ $$a\asymp_sb :\rightarrow\exists n\in\Bbb Z(as^n=b)$$ where $a^n$ is defined as follow 1)$a^0=e$ 2)$a^{n+1}=aa^n$ ...
15
votes
6answers
278 views

Mathematical trivia (i.e. collections of anecdotes and miscellaneous (recreational) mathematics)

Can you suggest some books on mathematical trivia? I use the word "trivia" with a double meaning in this case: curious anecdotes that enlighten what the real life of mathematicians is like (like ...
5
votes
3answers
406 views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
7
votes
3answers
340 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
3
votes
1answer
98 views

Except First year Abstract Algebra and commutative Algebra, what else do i need to start read Algebraic Geometry text?

Except First year Abstract Algebra and commutative Algebra text, what else do i need to read before start read Algebraic Geometry texts? I am refer to the beginning texts: "Algebraic geometry an ...
3
votes
1answer
61 views

What is a list of book that i need to read as a prerequisite before start reading “lectures of logic and set theory vol.1 by George Tourlakas”?

What is a list of formal textbook that i need to impose myself to read as a prerequisite before start reading a book called lectures of logic and set theory vol.1 by George Tourlakas? That book is ...
74
votes
22answers
2k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
4
votes
4answers
209 views

Good books written by great mathematicians

I read many of Richard Fenynman's books and I found them both very entertaining and moving, showing the human side of a brilliant scientific mind. I recently read also a collection of P.A.M. Dirac's ...
2
votes
2answers
151 views

Mathematicians average in student life but later became significant

What are the examples of mathematicians who were below the average in their student life (say, upto university level but it may be less) but later in life became significant mathematicians. Up until ...
0
votes
4answers
51 views

Recommendation for free graph plotter that can produce beautiful graphs

Can anyone please recommend a good free graph plotter that I can download. I am looking for a program that can produce neat looking graphs with all the axes, grids and that can plot many different ...
6
votes
3answers
265 views

Mathematical results that were generally accepted but later proven wrong?

I am giving a presentation on mathematical results that were widely accepted for a period of time and then later proven wrong, or vice versa. This talk is geared towards undergraduates who are likely ...
3
votes
2answers
44 views

Applications of Baire's Threom [duplicate]

In a lecture on Baire's Theorem (for complete metric spaces), I gave, for a rather advanced undergraduate class in Real Analysis (covering the theory of metric spaces and elements of general ...
2
votes
1answer
39 views

Other Useful Series Tests

So after taking calculus II, or maybe a first course in analysis, everyone learns a few series tests. They learn 1) Divergence Tests 2) Integral Test (from which we deduce things like $p$-series. ...
32
votes
1answer
686 views

What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?

I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract ...
2
votes
2answers
131 views

Introductory example(s) of a functor that is full but not faithful

What is your favourite example to offer real beginners of a functor which is full but not faithful?
0
votes
0answers
62 views

Favourite proofs by induction?

I am searching for nice proofs by induction, that can be used to teach. I remember this example, that my analysis professor presented to us in first semester and I am searching for more such easily ...
0
votes
0answers
11 views

Accessible resources to learn about bicharacteristic strips

I'm taking an introductory course in PDEs and, once seen the method of characteristics, the professor briefly talked about bicharacteristic strips and micro-local analysis. I'd hate to pass by such a ...
9
votes
2answers
310 views

How to show that five points in ℝ³ are cospherical?

There are many conditions equivalent to the cocircularity of four points on a plane, however i could not find any such lists for the three-dimensional analog. When do five points in three-dimensional ...
33
votes
18answers
2k views

Suggestion for Math Movies

I am interested in Math movies which inspire and motivate. I know about A Beautiful Mind, Good Will Hunting, and Pi. Are there any others someone can suggest?
1
vote
0answers
73 views

Important integral inequalities list.

What are the most important and usefull integral inequalities? I know Chebyshev and Schwarz. Google search provides very few results, mathworld doesn't provide a list.
2
votes
1answer
32 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
31
votes
5answers
2k views

“Stick it to the man!” Mathematical discoveries that resulted in persecution.

As the old story goes, Pythagoras and his followers were adamant that all numbers were rational, until Hippasus came along and proved that $\sqrt{2}$ (the length of the diagonal of the unit square) is ...
2
votes
4answers
190 views

Theorems in number theory whose first proofs were long and difficult

What are the examples of important theorems of number theory that has been shown to have surprisingly simple proofs though their first demonstration wasn't at all simple enough. Now simple proof is an ...
6
votes
2answers
109 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
8
votes
1answer
151 views

Reference request: books that describe application of physical reasoning to mathematical problems

I am searching for more books like Uspenski's Some applications of mechanics to mathematics and Levi's The Mathematical Mechanic. In other words, I am looking for books that show interesting and ...
1
vote
7answers
183 views

Text books on computability

I collected the following "top eight" text books on computability (in alphabetical order): Boolos et al., Computability and Logic Cooper, Computability Theory Davis, Computability and unsolvability ...
10
votes
2answers
585 views

Abstract algebra book with real life applications

Is there an abstract algebra book that emphasizes the applications to "real world" problems? Update: By real world, I mean mostly related to physics or other sciences. But references to coding theory ...
4
votes
1answer
103 views

The best of Martin Gardner…

Martin Gardner's 100th Birthday is just about to come and I am a huge fan of his books as well as his puzzles and games . I personally loved his puzzles like the "Reversed Trousers" which said ...
2
votes
0answers
99 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
9
votes
13answers
521 views

Mathematics and literature

Are there novels (or other kinds of books) that include substantial references to topics and ideas closely related to mathematics (even if there are no explicit references to theorems, proofs, ...)?