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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1answer
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Indefinite Integral challenge problems

This year I am going to participate in an olympiad of indefinite integrals. The level is very high, I would like to know some (hard, olympiad) Indefinite integrals challenge problems Note: Here is ...
6
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3answers
65 views

Tricks for quickly reading off the eigenvalues of a matrix

I noticed that some mathematicians have an uncanny ability to identify the eigenvalues of matrices without doing much in the way of computation. For instance, one might notice that all the rows have ...
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17answers
2k views

List of Local to Global principles

What are some of the local to global principles in different areas of mathematics?
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12answers
12k views

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...
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2answers
112 views

Lists of the first fundamental group of spaces. [closed]

Here are some list to start with $$\begin{array}{|c|c|c|} \hline \mbox{Space}(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& ...
76
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25answers
12k views

Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
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3answers
6k 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 ...
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9answers
2k views

Elementary Papers at ArXiv

Inspired by this question, at MO i am asking this question. Can anyone list some elementary articles at ArXiv which can be understood by High-School/Undergrad Students. I am asking this because, i ...
6
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4answers
240 views

'Almost rational' integrals with no known closed form?

I recently stumbled upon an 'almost rational' integral, namely: $$\int_0^{\pi/2} x \frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx=0.231231222\dots \approx 0.231231231\dots= ...
218
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65answers
50k 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 ...
10
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3answers
159 views

Fun, interesting, slightly advanced books

I came across a really interesting thread in the Internet where the author was asking for fun, but serious Maths book recommendations. I saw plenty of excellent books being recommended there and ...
27
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1answer
373 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
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6answers
3k views

What are Some Examples of “Non-Logical Theorems” Proven by Logic?

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
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0answers
140 views

Separating Heavier from the Lighter Balls

This was posted Here and received a good answer, solving the general questions in either $n$ or $n+1$ moves, which is by just half a move on average "less good" than my manual solutions here. ...
27
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20answers
972 views

What are some surprising appearances of $e$?

I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of ...
357
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34answers
22k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
2
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1answer
60 views

Classical results of Algebraic Geometry using cohomology.

I am looking for classical results of Algebraic Geometry that can be proved using cohomology. For example, Riemann-Roch Theorem and Bezout Theorem admits short proofs (providing that you know enough ...
5
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1answer
85 views

List of matrix properties which are preserved after a change of basis

Lately I encountered such a problem. Which of the properties of matrices are preserved after a change of basis ? (orthogonal basis and square matrix are preferred in the first place) Maybe it is a ...
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1answer
27 views

Comprehensive handbook of formulas

I'm looking for comprehensive, free and online handbook of mathematical formulas, in the form of pdf file. I mainly aim at undergraduate mathematics. Do you know any link? Thanks.
5
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2answers
78 views

Book about intuition behind Lebesgue measure

I recently completed a course in Real analysis covering Lebesgue and Borel measure, Fourier series, $L^p$ spaces and such. I can solve problems in these topics but am afraid that I do not truly ...
3
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3answers
233 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
20
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1answer
897 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
1
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1answer
77 views

Advanced examples of categories

I'm looking for some example of categories which requires some effort to prove that it is a category (For example it is straightforward to prove that $\mathbf{Set}$ is a category, I don't want that ...
43
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13answers
4k 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 ...
90
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24answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
2
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3answers
295 views

How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
67
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30answers
31k views

Best Maths Books for Non-Mathematicians [closed]

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
0
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1answer
70 views

Books on graphical models

I looked at Koller & Friedman's Probabilistic Graphical Models, but their use of non-standard notation is prompting me to see if there is anything else out there. I'd like to find an introductory ...
104
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22answers
11k 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 ...
0
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1answer
16 views

Calculating the amount of times a binary search could run (worse case) without a calculator/calculating base 2 logs without a calculator.

Ok so I had a question on a test that I had to do without a calculator. And I can not figure out how in the world I am supposed to do it without a calculator. The question asked to find how many ...
27
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6answers
865 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 ...
115
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10answers
9k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
13
votes
9answers
860 views

Elevator pitch for a (sub)field of maths?

When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of ...
1
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1answer
41 views

What are some pairs of mathematically-important functions that differ only at a few points?

Examples would include things like $$f(x, y) = \begin{cases} x^y & \text{ if } (x, y) \neq 0 \\ 0 & \text{ else} \end{cases}$$ versus $$g(x, y) = \begin{cases} x^y & \text{ if } (x, y) ...
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3answers
623 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
0
votes
1answer
627 views

How many ways to find the center of an inscribed circle?

I want to find the coordinates of center of the inscribed circle triangle $ABC$, where $A(-274,-253)$, $B(-1,7)$, $C(14,7)$. I tried First way. We have $c = AB=377$, $a = BC=15$, $b = AC=388$. Let ...
708
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53answers
419k 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 ...
26
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9answers
40k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
91
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5answers
4k 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 ...
1
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1answer
32 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
91
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24answers
73k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
621
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160answers
39k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
14
votes
5answers
214 views

Is there a property in $\mathbb{N}$ that we know some number must satisfy but don't know which one?

I have two questions. $(1.)$ Is there a property of the natural numbers such that we know at least one number satisfies it but we don't know which one? Even more, $(2.)$ Is there a property ...
103
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34answers
17k 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 ...
59
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15answers
2k views

Unconventional mathematics books

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it represents many of Euclid's proof as ...
42
votes
20answers
3k views

Which mathematicians have influenced you the most? [closed]

This question is lifted from Mathoverflow.. I feel it belongs here too. There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, ...
3
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1answer
44 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
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2answers
131 views

$a^x+b^x=c^x$ in geometry

The Pythagorean theorem. Let $A$, $C$, $B$ be three points on a line in this order, and let $D$ be another point, such that $\angle ADC =\angle CDB = 60^\circ$. Let $a=AD$, $b=BD$, $c=CD$. Then, ...
42
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10answers
12k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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4answers
598 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that ...