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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2
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0answers
88 views

Comprehensive Linear Algebra Text

Occasionally I come across a fact from linear algebra that I have not seen before. These facts are often obscured in search engines by either introductory texts or unrelated papers, and it is ...
13
votes
7answers
157 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
2
votes
1answer
94 views

Challenging problems in algebra (book recommendation)

Could you suggest me a book/web page where I can find challenging/hard problems in algebra (possibly with solutions) for an undergraduate student (groups, rings, fields, Galois theory)? Thanks in ...
6
votes
1answer
57 views

Books that start with questions? [closed]

Does anyone know of any books that start with a relevant question, study it from different perspectives and then show some mathematics? I would appreciate it if you posted any that you know of, ...
4
votes
0answers
49 views

What are some other examples of this phenomenon: if $S$ is a finite set, then all possible total orderings of $S$ are isomorphic (as posets).

Finite sets have the amazing property that if $S$ is a finite set, then all possible total orderings of $S$ are isomorphic (as posets). Said another way: finite totally-ordered sets that are ...
26
votes
6answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
0
votes
4answers
82 views

Easy proof of $\mathcal{P}(\mathbb{Q})$ is uncountable [Big list]

I'm looking for a easy proof of uncountability of $\mathcal P(\mathbb Q)$. I'll contribute with this: Let $\mathcal{P}(A)$ denote the power set of $A$, since ...
14
votes
10answers
1k views

Math problems that are impossible to solve [closed]

I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering ...
1
vote
0answers
47 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
-1
votes
0answers
7 views

Difference between the (Euclidean) hyperboloid and the (Hyperbolic) hyperboloid model.

I am getting completely confused on the differences and similarities between the (Euclidean) Hyperboloid and the (Hyperbolic) Hyperboloid Model and it looks like some people just mixthem upo ...
3
votes
1answer
56 views

Bounds for $n$-th prime

In this page I have found that the bounds for $n$-th prime is given by, $$n(\ln n+\ln \ln n)>p_n>n(\ln n+\ln \ln n-1)$$ for all $n\ge6$. Are there even stronger bounds for the $n$-th prime? If ...
11
votes
4answers
338 views

Elementary problems that would've been hard for past mathematicians, but are easy to solve today? [closed]

I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous ...
4
votes
0answers
85 views

Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
6
votes
5answers
211 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
2
votes
1answer
97 views

Amazing integrals and how is solved it [closed]

There a lot of integrals, however many people solved it in different ways, we can find interesting integrals in Table of Integrals, Series, and Products. I wonder What is the most exciting integral ...
0
votes
0answers
59 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
2
votes
0answers
31 views

Perimeters Areas and Volumes

I have to write an article for a school magazine. I thought it is better to choose a simple topic like Perimeter, Area and Volume. I am looking for historical fact and surprising facts about ...
4
votes
0answers
75 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
2
votes
0answers
25 views

How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
1
vote
2answers
44 views

Representative Pedagogical Examples of Groups, Real Functions, Modules, etc.

In the preface of Munkres's Topology, he writes, Fortunately, one does not need too many counterexamples for a first course; there is a fairly short list that will suffice for most purposes. Let ...
3
votes
0answers
104 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
0
votes
0answers
40 views

Looking for examples on Jordan Form

I am not asking a very specific question, but rather I am looking for any good examples that illustrate the following; $\mathbf{Theorem}: $ Let $T: V \to V$ be a linear operator with characteristic ...
3
votes
0answers
81 views

Funny translations of mathematical words [closed]

As already noticed in this question there are some mathematical words that literally translated from a language to english (or from english to this language) means something totally different. A few ...
3
votes
2answers
52 views

What other classes of commutative rings can be defined by requiring that $\{0\}$ is the only proper ideal satisfying some condition?

A field is just a commutative ring $R$ such that $\{0_R\}$ is the only proper ideal. Interestingly, there's a similar characterization of integral domains. Given a subset $A$ of $R$, let $A^\perp$ ...
18
votes
6answers
558 views

Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
9
votes
4answers
381 views

Big List of examples of recreational finite unbounded games

What are some examples of mathematical games that can take an unbounded amount of time (a.k.a. there are starting positions such that for any number $n$, there is a line of play taking $>n$ times) ...
3
votes
1answer
86 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
29
votes
4answers
1k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
41
votes
9answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
0
votes
1answer
33 views

Uniform convergence in series definitions of functions

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!
3
votes
1answer
76 views

Suplement books for calculus course?

I'm looking for books that could enhance the learning of calculus. At the moment, I have the following titles: Counterexamples in Calculus; Irresistible integrals; Inside calculus; The Calculus ...
6
votes
1answer
90 views

Surveys: problems, conjectures, and questions in some areas of nonlinear analysis

I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and ...
12
votes
6answers
720 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
11
votes
2answers
284 views

Open/publicly available textbooks worth their salt

I've been reading a bit about "open textbooks", i.e. textbooks made available for easy, online access. These can be nice for those without access to a great library, or who might not be willing to ...
6
votes
5answers
571 views

Counterexamples for “every linear map on an infinite dimensional complex vector space has an eigenvalue”

Every linear map on a finite dimensional complex vector space has an eigenvalue. Not so in the infinite case. I'm interested in nice counterexamples anyone might have. Here's one: Consider the ...
-1
votes
1answer
63 views

Does Analysis Appear on Putnam?

I was recently searching past Putnam problems, I believe I saw some sort of Real Analysis, but I am not sure. Is Real Analysis (Mathematical Analysis) a topic, which appears on the Putnam Exam? Which ...
0
votes
0answers
301 views

Ahlfors Complex Analysis solutions for reference [on hold]

I am a math major and currently I am self studying complex analysis from Ahlfors Complex Analysis - third edition. I prefer working on problems first by myself and then refer back to see if I am right ...
2
votes
0answers
51 views

GRE Mathematics Practice Exams

I will be taking the subject test in the near future. Can you recommend me some sources (online or print) from which I can find realistic practice exams? I would like to get my hands on as many ...
1
vote
2answers
122 views

Doing Michael Spivak's Exercises

I am doing Spivak's Calculus, and I find it EXTREMELY difficult. I usually ask questions here because I cannot do the problems on my own. How long should it take to do a Spivak problem? Is it ...
4
votes
1answer
54 views

What are the subjects an analytic number theorist must be well versed with after undergraduate studies?

I am a mathematics major and I aspire to be an analytic number theorist. In general, what are the subjects an analytic number theorist must be well versed with after undergraduate studies (i.e. in ...
15
votes
4answers
478 views

Books in the spirit of Problems and Theorems in Analysis by George Pólya and Gábor Szegő

In the Preface of the first German Edition of the book Problems and Theorems in Analysis by George Pólya and Gábor Szegő, one can read [emphasis mine] : The chief aim of this book, which we trust ...
1
vote
3answers
65 views

Different Types of Waves

I am making a basic 2D rigid body simulator as a hobby. It involves springs. Naturally, I need to render them. Rigid body simulators, such as Algodoo, render them simply like this Another (more ...
2
votes
1answer
105 views

Learning Olympiad Level Combinatorics

Combinatorics has always been my weakest point, I want to improve it. There are such problems like: "Five friends should give each other gifts. They have made a gift each, as they should give away ...
1
vote
0answers
58 views

What are some really cool problems that involve “least squares and Eigenvalue problems”?

I am required to find a research topic in this domain, so I'm really interested in finding out what kind of problems are covered in this domain, and how others are using these techniques to solve ...
1
vote
0answers
34 views

Other Diophantine problems that use a Pell equation

What Diophantine equations employ Pell equations in their solutions? A well-known example is the case of Pythagorean triples where the legs differ by 1, like, $$20^2+21^2 = 29^2$$ These are ...
1
vote
3answers
118 views

Higly axiomatic geometry book recomendation

Recently I have started dipping my toes in mathematical waters besides calculus,and with varying success I have started learning bit of something about "everything". But I have one issue,namely I can ...
1
vote
1answer
31 views

Different ways to prove convexity of quadratic form associated to rank 1 matrix

Let $v \in \Bbb R^n$, and $f:\Bbb R^n \to \Bbb R^n$ with $f(x)=\langle x,(vv^T)x\rangle$. Show that $f$ is convex. I'm looking for different approaches to solve this (rather simple) problem. ...
4
votes
3answers
158 views

Do you know any almost identities?

Recently, I've read an article about almost identities and was fascinated. Especially astonishing to me were for example $\frac{5\varphi e}{7\pi}=1.0000097$ and ...
49
votes
10answers
1k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
22
votes
9answers
1k views

Surprising applications of topology [closed]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...