Questions asking for a "big list" of examples, illustrations, etc. Please do not ask too many of these. Please do not use this as the only tag for a question.

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5
votes
0answers
164 views

What is the best Mathematical Insight you have had? - PLEASE MOVE TO META [closed]

I've used this site a lot but am new to the actual forum. Basically, I am wondering if we could collect a list of mathematical insights / explanations / neat proofs etc. that people on this forum have ...
0
votes
1answer
36 views

In how many different ways can this problem be solved?

I have a math problem. In many different ways can this problem be solved? Here is the problem: $$y''-y'-2y=0, \\ y(0)=1 \\ y'(0)=0$$ I have already found $5$ ways: $(1):$ Characteristic equation ...
5
votes
2answers
139 views

Why the $\log$ is so special?

When I first learn about the logarithm function $\log$ or $\ln$. My professor said that $\log x$ is a function that when we derive we get the inverse function $1/x$. This $\log$ becomes very popular ...
26
votes
6answers
3k views

Tell me problems that can trick you

I am looking for problems that can easily lead the solver down the wrong path. For example take a circle and pick $N$, where $N>1$, points along its circumference and draw all the straight lines ...
84
votes
20answers
15k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
2
votes
2answers
167 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
0
votes
1answer
52 views

Overview of game theory

I have a good high school math background, and I am interested in game theory, so I wanted to know something more about it, but I found very technical things or wikipedia. I am looking for something ...
0
votes
0answers
72 views

What are $\Gamma$-semigroups?

I have some problems with $\Gamma$-Semigroups, the definition that I've found is A $\Gamma$-Semigroup is a pair $(M,\Gamma)$ defined as follow If $x,y$ and $z$ are in $M$ and $\alpha$ and ...
452
votes
45answers
303k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
5
votes
1answer
55 views

Inequalities that show if a distribution decays slowly

Often, one is often interested in theorems/inequalities of the following kind: Let $X$ be a random variable then the probability that $X$ is close to typically $\mu$ (or larger than some constant) is ...
10
votes
1answer
229 views

Theorems in algebraic geometry which have been proved only by using cohomology

There are many theorems in algebraic geometry which were proved using cohomology. I would like to know examples of such theorems which have been proved only by using cohomology. In other words, those ...
2
votes
2answers
229 views

Simple & Intuitive Statements that are Difficult to Prove

Looking through the webcomic, I came across one of my favorite comics: (from Saturday Morning Breakfast Cereal) It seems that people have an ongoing interest in results in mathematics that are ...
4
votes
3answers
422 views

Most inspirational mathematical books [closed]

I would like to know which books on mathematics (from university texts to divulgative pop-math books) inspired you the most. My choice is Spivak's Calculus, which is, IMHO one of the most ...
8
votes
4answers
1k views

Interesting mathematical problems for 1st year university students [closed]

Can you explain some mathematical problems that you find the most interesting (NB: the problem must be accessible to a 1st year university student: that is, a problem for which there is an elegant ...
5
votes
1answer
173 views

Which mathematical game or puzzle did you invent?

A couple of weeks ago, a friend of mine showed me a extension of a game we are all familiar with that he was working on. The game we know is called Tic-Tac-Toe, and he was working on his own version ...
23
votes
7answers
644 views

“Here's a cool problem”: a collection of short questions with clever solutions

This game will be familiar to many mathematicians, and it is always good fun to play. I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one ...
-1
votes
1answer
79 views

A big list of examples that a power of a prime ideal is not primary in an algebra of finite type over a field

Let $k$ be a field. Let $A$ be an integral domain which is a $k$-algebra of finite type. I would like to know examples that a power of prime ideal of $A$ is not primary. The more example, the better. ...
10
votes
2answers
614 views

Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
0
votes
1answer
126 views

Examples of non-trivial closed subschemes of a complete non-projective non-singular variety

Let $k$ be an algebraically closed field. A variety over $k$ is a separated integral scheme of finite type over $k$. Let $V$ be a complete non-projective non-singular variety over $k$. Let $Z$ be a ...
5
votes
2answers
141 views

How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
3
votes
3answers
227 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
2
votes
0answers
30 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
0
votes
2answers
55 views

What is list of common integral that have no closed form?

What is list of common integral that have no closed form? It's diffucult for me to google it for some reason.
5
votes
1answer
104 views

Why Did You Specialize in X?

For those of you who are researchers or graduate students, why did you choose to specialize in the field of mathematics X (as opposed to some other field Y)? Is it because you think X is important, ...
6
votes
3answers
209 views

Definitions which should be propositions/theorems

I am asking for a list of concepts which some sources present as definitions whereas other sources pose them as propositions/theorems. For example, most abstract algebra books will define a group ...
92
votes
32answers
16k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
4
votes
1answer
114 views

What are different notations used by mathematicians and physicists?

One can find many cases that mathematicians and physicists use different notations for the same concepts. Here is a few cases I find. Inner product of vectors: Mathematicians use $(a,b)$ or ...
4
votes
4answers
215 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
19
votes
8answers
651 views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
3
votes
2answers
218 views

Hard problems book in linear algebra

Could you suggest me a book where I can find hard problems in Linear Algebra for an undergraduate student? Thanks in advance.
2
votes
0answers
390 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
4
votes
1answer
237 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
81
votes
18answers
8k views

How do you describe your mathematical research in layman's terms?

"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure ...
7
votes
0answers
219 views

Most famous competition problems? [closed]

When I've attended math competition discussions, I've often heard people remark "oh, this is a famous problem" or say that it's similar to one. Most of them I've actually never heard of before. ...
7
votes
2answers
145 views

Ways to induce a topology on power set?

In this question, two potential topologies were proposed for the power set of a set $X$ with a topology $\mathcal T$: one comprised of all sets of subsets of $X$ whose union was $\mathcal T$-open, one ...
0
votes
1answer
75 views

Creative easy combinatorics problems. [closed]

I would like cool problems of the following style: how many marbles need to be taken out of a jar to guarantee we have one of each color? I need some cool problems for some classes I want to give to ...
6
votes
3answers
209 views

What are some properties that imply that a group must be the trivial group?

In the problem posed in this question of mine we want to show that a particular group is both perfect and solvable, and therefore trivial, and this turns out to be useful in proving the result. What ...
15
votes
3answers
352 views

which exact integration techniques belong in a first year calculus/analysis course?

At our university we are now discussing changes to the course contents and there is some heated discussion regarding integration in the first year calculus courses. Currently, the techniques of exact ...
7
votes
1answer
110 views

What is your favorite group? [closed]

I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please ...
2
votes
2answers
302 views

When are analytical solutions preferred over numerical solutions in practical problems?

In most engineering or applied math papers that I read, the authors seem to obtain solutions to say, a system of differential equations, using numerical methods, rather than analytical techniques. ...
0
votes
0answers
102 views

What are some great Bachelor's Project subjects in the field of Mathematical Optimization Theory?

Currently, I ought to pick a subject for my third year mathematics bachelor's thesis. I would like to research something in the field of mathematical optimization theory. I have a background in basic ...
0
votes
1answer
177 views

Most “beautiful” presentations of the basic proofs for vector spaces?

I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly ...
2
votes
1answer
577 views

Lecture Notes in Real Analysis

I understand that this question was partially addressed here but I would like to have a question dedicated to just real analysis. I am looking for both elementary real analysis (advanced calculus type ...
5
votes
2answers
137 views

Theorems that have proofs from the outside of the original field of math

I would like to know more examples of theorems, which "belong to one field of math", but their proofs are from the "outside of the field". I am mostly interested in proofs that are not too long ...
6
votes
2answers
160 views

Sources of Elementary Number Theory Problems

I am looking for sources of interesting and challenging problems that would suitably accompany an honors level introductory number theory course. What are some good sources for interesting elementary ...
17
votes
3answers
302 views

Results in graph theory proved using other areas of math, and vice versa

I'm curious about learning graph theory, as it seems to pop up in some unexpected places. In order to get a partial feel for the subject, I was wondering if anyone could point me to some survey ...
7
votes
1answer
353 views

“All math is useful eventually”

We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in ...
2
votes
2answers
77 views

when is a ring a free module over a subring?

Let $S \subset R$ be rings, $S$ not necessarily an ideal of $R$, and $S \neq R$. Is there anything that can be said about when $R$ is free as an $S$-module?
2
votes
1answer
180 views

Statements equivalent to the Axiom of Choice

The Axiom of Choice reads: The product of a collection of non-empty sets is non-empty. As you know well, this axiom is equivalent to many other statements. A few examples (probably the most ...
11
votes
1answer
387 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...