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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1answer
109 views

Crazy Set Theory Analogies

I think the following analogies are too interesting to be ignored: Union = Least Common Multiple If $G_1,...,G_n$ denote a number of sets of points (either linear or in any number of dimensions), ...
1
vote
1answer
64 views

Seemingly hard integrals which are made easy via differentiation under the integral sign a.k.a Feynman Integration

I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this ...
-3
votes
0answers
30 views

IB Math Help: grade Nine [on hold]

Does anybody who has been or is in grade 9 IB program know what sort of material they put on the entrance tests. Just a brief list of the math topics covered in the examination. Only answer if you are ...
3
votes
5answers
92 views

Classic Counting Problems

Does anyone have some good, classic, counting problems? I want things that are interesting, as well as instructive- more than just compute the number of way to get a flush, etc. (Not that those aren't ...
0
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0answers
29 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
0
votes
0answers
112 views

“Deep” maths books in certain subjects [closed]

I would like a suggestion on the 'deepest' books in Calculus and analysis (something along the lines of Rudin's) Linear algebra Abstract algebra Geometry (and topology); (even something along the ...
1
vote
2answers
130 views

Mathematical statements that cannot be proved or disproved [on hold]

I've recently been reading about the continuum hypothesis and am fascinated by the fact that it cannot be proved or disproved, despite the fact that the statement itself is either true or false. What ...
0
votes
0answers
61 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
1
vote
1answer
50 views

Identities involving adjoint action

I'm looking for list of identities involving adjoint action $\mathrm{ad}_A X = [A,X] = AX - XA$. For example, it can be easily shown that: \begin{equation} e^{\mathrm{ad}_A} X = e^A X e^{-A} ...
9
votes
4answers
341 views
1
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1answer
39 views

Interactions between geometry and graph theory.

I'm looking for some nice theories or just exercises, with both geometrical aspects and graph theoretics aspects. Example may include for instance the 4-color theorem or Euler characteristics, maybe ...
1
vote
1answer
50 views

Other Interesting solutions to $a=bq+r$? [closed]

The division algorithm says $a=bq+r$, with $r$ between $0$ and $b$. Are there interesting restrictions on $r$ using number-theoretic properties that make the equation $a=bq+r$ hold, or hold with ...
4
votes
1answer
55 views

References for mathematical theory of summability of divergent series

Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign ...
0
votes
2answers
61 views

Important applications of the Uniform Boundedness Principle

There's like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, ...
4
votes
2answers
139 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
3
votes
5answers
294 views

Real life examples of order relations.

It's easy to find examples of equivalence relations (for example, A shares room with B), but I can't seem to find a real life example of an order relation (that is, a relation that's reflexive, ...
2
votes
1answer
73 views

Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
3
votes
0answers
72 views

Is there any visual animation to show the basic concept of algebraic geometry? [closed]

Is there any visual animation to show the basic concept of algebraic geometry? There are rarely pictures in textbooks, so are there any animation to show basic but important concepts?
2
votes
2answers
142 views

Daily exercises to speed up my mental calculations?

When I was a kid in school my father prevented me from using a calculator when solving my math homeworks. However at that time I was not convinced as of why not to use such a useful tool! So I kept on ...
3
votes
4answers
114 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
1
vote
1answer
52 views

What are the problems that you tried to find their solutions and you did not know that it is impossible?

Tell us your story about Mathematics. Have you dream one day to do a big contribution in Mathematics because you are curious and love challenges. What are things that you tried to prove which then ...
1
vote
0answers
65 views

Things you've believed for a long time were true, but are false in reality [duplicate]

Do you have any things (mathematical statements, statements about mathematics) you've believed for a long time were true, but now with enough mathematical knowledge you realize were wrong? For ...
13
votes
1answer
478 views
+100

Big-Daddy-Conjectures and Hierarchy of Mathematical Conjectures

I am interested in the Hierarchy and Connections between various different open problems in Mathematics, and the most general conjectures in various fields of Mathematics. Examples of Hierachy ...
0
votes
1answer
29 views

What are the purposes for all of the distributions?

I want a list of all of the main distributions and their application: Example entry: Poisson$(\lambda)$ - Models number of events in a given amount of time or space. Thank you, community answer ...
0
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3answers
115 views

What are other unexpected results of integration?

I have integral of $\dfrac{1}{t^2 + 1}$ and integral of $\dfrac{t}{t^2 + 1}$ whose output is $\arctan(t)$ and $\dfrac12\ln(t^2 + 1)$ respectively. Are there any similar unexpected results when we ...
5
votes
0answers
69 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
1
vote
3answers
151 views

How many different definitions of $e$ are there?

It seems as though, in my analysis and calculus courses, in particular, a common cop-out when asked to prove an identity involving $e$, is the phrase "it's true by definition". So, I'm trying to find ...
5
votes
1answer
102 views

Category theoretic approaches to Riemann Hypothesis?

I was wondering if there has been any category theoretic advancements in the study of the Riemann Hypothesis and the theory surrounding it? This question is meant in the same vein as these ...
0
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0answers
19 views

Reference request for papers proving the existence of a special function satisfying the following conditions

Previously in this site it has been proved that there exists at least one prime between $c_n$ and $n$ where $c_n$ denotes the $n$-th composite (see the question Prove that there exists an $m$ such ...
46
votes
46answers
4k views

What was the book that opened your mind to the beauty of mathematics?

Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But ...
0
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0answers
46 views

Has this property for algebraic structures got a name?

Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$) $a*A:=\{a\}*A$and $A*a:=A*\{a\}$, ...
9
votes
3answers
257 views

Book series like AMS' Student Mathematical Library?

I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
14
votes
3answers
329 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
1
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0answers
23 views

What [precisely] is special, algebraically, about fundamental Pell solutions?

I'm asking for a list of algebraic identities which "uniquely identify" (or nearly so) the fundamental solution (or those immediately around it) of a Pell equation. As a concrete [numerical] example, ...
3
votes
2answers
74 views

Examples of misleading notation that gives correct results

A lot of the time, in maths, especially when I'm trying to remember a formula, I'm taught to remember it in a way that is not notationally correct, but produces the right result. e.g. when learning ...
3
votes
2answers
122 views

What jobs in Mathematics are always in demand, and are deeply Mathematically specialised or greatly general?

I am wondering what jobs in the field of Mathematics are (seemingly) always in demand. I am also wondering what jobs there are that are (once again seemingly) greatly Mathematically demanding in ...
141
votes
62answers
32k views

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved ...
2
votes
1answer
41 views

Good resource for learning braid theory?

I recently heard about braid theory and read the Wikipedia article on it, and it seems really beautiful. What is a good resource for learning more about it? I have a background in mathematics at the ...
9
votes
0answers
183 views

Paradoxical models of $\sf ZF$ without choice [closed]

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold ...
16
votes
6answers
1k views

Best Math books or apps for adults to learn math from the beginning

I lost a possible job because I didn't know how to multiply and subtract negative valued integers. I also don't know how fraction manipulation works. What reference books can I read that can help for ...
0
votes
1answer
38 views

Theorems for Pascal's triangle

The mathematician Donald Knuth (b. 1938) once indicated that there are so many relations and patterns in Pascal's triangle that when someone finds a new identity, there aren't many people who get ...
9
votes
2answers
174 views

Examples of theorems that haven't been proven without AC in practice but can be proven without it in principle

It is possible to prove theorems of the form "if $\phi$ is provable in ZFC, then $\phi$ is provable in ZF". For example, let $\phi$ be a statement that is absolute between $V$ and $L$. If $\phi$ were ...
0
votes
1answer
40 views

Exhaustive list of recreational mathematical concepts

There are many simple yet elegant, addictive and entertaining mathematical concepts. For example, drinker paradox, pigeon hole principle, Monty Hall problem, Hilbert's paradox of the Grand Hotel, etc. ...
1
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2answers
53 views

When trees are the answer: what is the question?

For which optimization problems are (abstract) trees the best solution? E.g. binary search trees are somehow optimal data structures for quick search. But why for example do botanic trees grow as ...
0
votes
0answers
23 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
16
votes
3answers
887 views

An equation that generates a beautiful or unique shape for motivating students in mathematics

Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet: $$ \large\color{blue}{ ...
0
votes
1answer
17 views

List of Bounds of $n$-th composite

I am looking for a list of all the bounds on $c_n$, the $n$-th composite. There is a trivial bound $2n \geq c_n >n$ $\forall n \geq 5$. But I am looking for bounds stronger than this. I have ...
1
vote
1answer
45 views

Interesting Identities in First-Order Logic

Are there more identities of this sort (http://en.wikipedia.org/wiki/First-order_logic#Provable_identities) that are interesting/non-trivial? It seems that most further work in first-order logic is ...
14
votes
3answers
338 views

What are the uses of “squeezing”?

Off hand, the uses of "squeezing" that I can think of are: showing that $\lim_{x\to0}\dfrac{\sin x}x = 1$, which is then used in finding derivatives (PS: I've just remembered this item showing ...
6
votes
4answers
148 views

The Power of Dimensional Analysis

What are some illustrative examples of dimensional analysis at work? Especially, where can it be used to significantly reduce the difficultly of a computation?