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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What are some topics in undergraduate mathematics to write about? [closed]

As a high school student, I will soon be going on to do the IB(college equivalent). As part of the IB curriculum we are expected to write an Extended Essay(EE) that counts towards our grade. For the ...
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4 votes
2 answers
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Build a "rich" first-order logic within a given category

I would like to know a mathematical framework with an internal logic where isomorphic objects can be considered equal. For example, consider the rationals $\mathbb{Q}$. With this set we can construct ...
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A Collection of Bogus Proofs

Hello M.S.E. people, This question is just for fun, don't take it seriously :). We have all encountered Bogus Proofs, which seem logical and reasonable, but they prove some claims which are completely ...
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1 vote
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Strong ("second principle") mathematical induction: non-trivial examples

The fundamental theorem of arithmetic (FTA) can be proved using the following: if a statement is true for $n=1$, and its truth for $n=1,2,\ldots,k$ implies its truth for $n=k+1$, then it is true for ...
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Obscure Results and Areas of Mathematics

My degree is in economics although I did relevant mathematics in college and used algebraic topology in my thesis. Over the Pandemic I've been reading a lot of mathematics that I didn't get a chance ...
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7 votes
3 answers
151 views

Mathematics hiding in plain sight

What are some basic math facts (say, secondary or early undergraduate level) that somehow went unnoticed by you for a long time, and when you realized they made you wonder how you could have missed ...
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1 vote
0 answers
30 views

Good articles/books to learn about tensors / tensor products?

Currently a mathematics graduate student at a state university. I'm looking for articles or books (preferably free PDF links) that nicely explain tensors and tensor products at a graduate student ...
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4 votes
0 answers
53 views

Examples of nontrivial proofs by cases in which only one case is ever realized

For teaching reasons, I'm looking for examples of proofs that use a nontrivial case breakdown in which only one case is ever realized, and yet it is very hard to prove which case is the "real&...
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3 votes
1 answer
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Mathematicians endorsing platonism -- examples?

Take platonism to be the view that there are abstract mathematical objects which exist independently of us as mathematicians and our language, thought, and practices. Looking at the Stanford ...
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7 votes
4 answers
298 views

Examples of non-elementary integrals, but whose definite integral IS solvable with power series.

In a high-school level calculus course you learn about Taylor series and some basic integration techniques. In my experience, most definite-integration exercises boiled down to finding an ...
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primorial $\pm1=$ prime (sometimes)

I know there are a few questions about this sort of thing already, so excuse me if this is a duplicate. I noticed something interesting when playing around with primorials and related quantities. I ...
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1 vote
2 answers
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Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs?

Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs? Examples of the scalar versions: 1st order: $\dot{x} = F(x)$ 2nd order: $\ddot{x} = F(x,\dot{x})$ I have ...
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1 vote
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What are some cool operations like Tetration? [closed]

I am aware of the large lists already present for example : Surprising identities / equations and Funny identities. However my question is different as I am looking for special kinds of non-standard ...
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-2 votes
1 answer
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What's an example of a theorem with multiple ways to prove it? [closed]

Please give a simple example that incorporates basic math, like basic arithmetic. Please include the theorem and the multiple proofs (can just mention them, or include the entire proofs).
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3 votes
0 answers
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Folklore Theorems in Group Theory.

This search for "folklore" in the group-theory tag suggests that this question is new to MSE. I am aware that this might be too broad. If it is, I'm sorry. I have included the big-list tag ...
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3 votes
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Examples of Cauchy complete ordered fields that are not $\mathbb{R}$?

According to this post, it is not true that Cauchy completeness (every Cauchy sequence has a limit) and Dedekind completeness (every nonempty set that is bounded above has a supremum) are equivalent ...
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3 votes
1 answer
99 views

What computer methods are used to quickly calculate the $\zeta$-function (if any)?

So I can think of how I could compute $\zeta(\sigma + it)$ in principle. We can take $\zeta(\sigma + it)$ for $\sigma>1$ by the usual $\sum_{n} n^{-(\sigma + it)}$ summation. We can use the ...
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4 votes
1 answer
105 views

What are some mathematical theorems which become unknown if you reject the axiom of choice?

I would like to see statements X such that: (1) We can prove X is true under ZFC. (2) We do not know whether X can be proven without the axiom of choice. For example, the statement "every vector ...
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5 votes
1 answer
113 views

Examples of Classical Approaches Being Potentially More Preferable than Modern Approaches

It is a well-known result that given a field $F$, the set of elements algebraic over $F$ is also a field. When presenting this result in class, my professor proved it two ways. What he called the &...
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1 vote
0 answers
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Is there any list of almost all subspaces of $\ell ^\infty$?

Is there any long list of subspaces of $\ell ^\infty$ (The set of real bounded sequences equipped with the supremum norm). This would help to find a great many counterexamples. Moreover, I wonder how ...
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5 votes
2 answers
161 views

Exotic Definitions of Groups

Inspired by this question I was wondering, whether there are alternative definitions of groups, namely ones different from the usual 4 axioms. I already suspected that the category theorists have one ...
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2 votes
1 answer
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Summation formulas for integral transforms other than the Fourier

It seems to me that the Fourier transform harbours multiple useful results that allow one to sum an infinite series. The Poisson Summation Formula and Parseval's theorem. Also, Plancherel's theorem ...
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0 votes
1 answer
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Examples of topological spaces with canonical bases with the following property (redivisibility)

Let us say that a family of sets $\mathcal{S}$ is redivisible if, for any $S_1, S_2\in \mathcal{S}$, the intersection $S_1\cap S_2$ is the union of finitely many sets from $\mathcal{S}$. Which (...
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7 votes
0 answers
241 views

Proofs of the Cayley-Hamilton Theorem [duplicate]

The idea of this post is for people to post different proofs of the Cayley-Hamilton Theorem. You can either try to post your own proof or give a reference. If you usse a reference, please give some ...
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2 votes
0 answers
107 views

What are (interesting) examples of theorems in Analysis in which $\mathbb R^n$ is the only exceptional case?

I can’t recall so many statements in Mathematical Analysis that hold for a certain class of possibly unbounded subsets of $\mathbb R^n$ (or $\mathbb C^n$) with the only exception being the whole space....
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2 answers
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Computing limits using integrals: examples

I am a highschool teacher and will teach integrals next term. Today I saw a video from blackpenredpen in which he computed the limit $\displaystyle \lim_{n \to +\infty}\left(\frac{n!}{n^n}\right)^{1/n}...
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Visualizing vanishing sets, online tools for algebraic geometry

Is there an online tool which can visualize algebraic sets such as $V(x+xy^2)$ or similar vanishing sets? More generally, are there some neat programs found online, which help getting a grasp of such ...
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4 votes
2 answers
204 views

Are there any non-constant "bump functions" in "closed form" whose Fourier Transforms are also in closed form?

I am looking for the simplest cases possible of one-variable bump functions $\in C_c^\infty$ [1] with known Fourier transforms in "closed form" (also the function itself). This means it can ...
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0 votes
0 answers
97 views

Visually intriguing unsolved problems which are easy to explain

I have come across a list of visual proofs which are wrong (Visually deceptive "proofs" which are mathematically wrong) visual proofs which are not wrong (Proofs without words) visually ...
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1 vote
0 answers
134 views

Goldbach-like and Collatz-like conjectures and theorems

I am looking for examples of conjectures and famously hard-to-prove theorems which can be stated in the form: For each natural number $n$, $P(n)$. where $P$ is a predicate of natural numbers that is ...
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  • 1,638
28 votes
9 answers
1k views

Various ways to calculate $\int \sin(x) \cos(x) \, \mathrm{d}x$

Consider the integral $$\mathcal{I} := \int \sin(x) \cos(x) \, \mathrm{d} x$$ $ \newcommand{\II}{\mathcal{I}} \newcommand{\d}{\mathrm{d}} $ This is one of my favorite basic integrals to think about as ...
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1 vote
0 answers
58 views

Fun topic for undergraduate seminar

I have already asked myself (and been asked) a couple of times what would be possible topics for undergraduate seminars. In such a seminar students are given passages of a book (or even paper) which ...
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1 vote
2 answers
95 views

Interesting applications of major results in complex analysis

I know of two examples of major theorems in complex analysis being used to answer problems that are easy to state but otherwise not so easy to prove: using the Maximum modulus principle applied to $\...
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2 votes
0 answers
104 views

Examples of invariance in plane geometry

In plane geometry, we often see that the locus of a particular geometric object remains invariant in all possible configurations of a set of points. For example, let $ABC$ be a triangle and $X,Y,Z$ be ...
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5 votes
4 answers
450 views

Examples of non-trivial exclusively irrational integrals?

One very famous integral is $$\int_{\mathbb{R}} \frac{\cos(x)}{x^2 +1} \, dx = \frac{\pi}{e} \tag{1}$$ as is shown in the answers to this question. I find this integral particularly interesting as ...
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3 votes
2 answers
135 views

Problems that can be shown by ”an epsilon of room”. [closed]

This might not be a question suitable here so apologies from posting it if so. In Terence Tao’s blog he refers to something called ”An epsilon of room” and this seems to be a helpful way to prove for ...
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1 vote
0 answers
36 views

Online Math Colloquia

One of the formative experiences of my undergraduate education in math was attending department math talks. These were usually aimed at general faculty members, so they went well over my head at the ...
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0 votes
3 answers
66 views

Important or interesting mathematical objects that are fixed points

I've observed that many interesting or important objects that arise in mathematics later turn out to be fixed points for some function (possibly in a way that is not obvious from their original ...
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1 vote
2 answers
92 views

Differently named concepts that turn out to be the same

(Apologies if this is a duplicate. Wasn't able to find a question with similar keywords) Motivation: Recently started reading Willard, General Topology. In one of the first exercises, he defines a ...
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3 votes
1 answer
95 views

Real life examples of ultrametrics or "the isosceles triangle principle"

Mathematical Background and Definitions: The distinguishing feature of an ultrametric is the "strong triangle inequality" i.e. for all $x,y,z$, $$d(x,y) \le \max(d(x,z), d(y,z)).$$ This ...
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4 votes
1 answer
89 views

Unexpected use of linearity of expectation with indicator random variable in problems

Can people suggest some problems (probability puzzle type) where the use of linearity of expectation together with indicator random variable is unexpected/hard to see but it makes problems much easier?...
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0 votes
2 answers
100 views

"Random" facts that math can prove (eg, there are at least 2 people on the planet with the exact same number of body hairs)

I just learned that you can prove mathematically that in the entire planet, there are at least 2 people with the exact same amount of body hairs. I was fascinated by this and I searched for more ...
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1 vote
0 answers
102 views

Books for a conceptual understanding of linear algebra?

I've been working Linear Algebra a lot recently; I've been using Schaum's Outlines for Linear Algebra to supplement my understanding. Unfortunately, as nice and organized and Schaum's tends to be, the ...
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0 votes
1 answer
80 views

Other Important Zeros? [closed]

The Riemann hypothesis conjectures that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part equal to $1/2.$ There are consequences and ...
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11 votes
6 answers
2k views

Problems with seemingly not enough information

Two of my favorite geometry problems are as follows: Consider two concentric circles with the property that a chord of the larger circle with length 20 is tangent to the inner circle. What is the ...
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1 vote
0 answers
94 views

List of mathematical portmanteaus

Students (including myself) are always fascinated when they learn words like "clopen" set for closed and open in topology, or "club" set for closed and unbounded in set theory. I ...
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4 votes
2 answers
236 views

What kinds of fields do exist that are non-perfect?

This question has already been asked twice in a similar manner (Non-Perfect Fields and Examples of fields which are not perfect). In both cases, the standard answer found in introductory textbooks was ...
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0 votes
0 answers
78 views

How many kinds of calculus are there? (With some notion of the derivative)

Name motif Single variable calculus Mathematical study of continuous change of single variable functions wiki Complex variable calculus Analysis of complex functionals wiki Calculus of variations ...
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0 votes
2 answers
137 views

About hereditary properties in abstract algebra [closed]

A hereditary property is a property of an object that is inherited by all of its subobjects where the meaning of sub-object depends on the context. Particularly the hereditary property is of great ...
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2 votes
1 answer
62 views

Books on Mathematical Games

I am looking for books on mathematical games that don't include probability. For example, books that cover the game of nims and other such stuff. Could I get some recommendations on it? I would prefer ...
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