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.

learn more… | top users | synonyms

23
votes
4answers
1k views

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of ...
2
votes
1answer
50 views

Applications of Nagata's Lemma

In the spirit of this MO question I would like to ask for applications of a (somewhat lesser known?) lemma. Lemma. (Nagata) Let $R$ be an atomic domain. TFAE: $R$ is a UFD There exists a ...
7
votes
0answers
72 views

Undergraduate group theory Aha facts [closed]

I am learning group theory. I found out that there are a lot of (more or less) simple facts that are completely trivial for working mathematicians (e.g. my professor), but that are nowhere explicitly ...
0
votes
0answers
17 views

Is there more RBF's kernel?

I am working on a sfde(stochastic fractional differential equation),I use some of Radial basis function to find solution.like this list ...
37
votes
10answers
1k views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
1
vote
0answers
42 views

Recommendations on visualizing basic linear algebra

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear ...
0
votes
0answers
57 views

Learning multivariable/vector calculus through guided discovery

I am asking this question as a question similar to what has been asked previously for other topics as well as math in general. But I'd like to ask for text references specifically in the domain of ...
5
votes
1answer
98 views

Unexpectedly uniformly continuous functions

The other day in a exam, I was given the following exercise: Given $f : [0,1] \to \mathbb{R}$ continuous and such that $f(0) = 0, f(1) = 1$, let $g : \mathbb{R} \to \mathbb{R}$ be $g(x) = [x] + ...
4
votes
2answers
176 views

Definitions of “linearity” across branches of mathematics or levels of math education

Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, ...
6
votes
2answers
64 views

Uses of vector spaces over $\mathbb Q$

I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$: Hilbert's third problem; and The Buckingham pi ...
7
votes
0answers
112 views

Using multiple integrals for tough single integrals

I'm just getting started on double integrals, and I recently saw the super cool way to use double integrals to arrive at $$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$ So, I am wondering if ...
1
vote
1answer
114 views

Humorous mathematical essays

Even though there are plenty examples of mathematical jokes, the mathematical literature is (in many cases) pretty dull. Nevertheless, examples exist in which an essay makes you smile with a nice pun ...
1
vote
0answers
44 views

Evaluating definite integrals using complex contour integrals

In most cases one only has to consider the complex function where we put $z$ instead $x$. For example, calculating the integral: $$\int_{0}^{\infty}\frac{dx}{x^{4}+1}$$ Here i just integrate the ...
10
votes
7answers
770 views

What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ...
2
votes
2answers
35 views

Books in the spirit of A. Cox “Primes of the Form $x^2+ny^2$”

David A. Cox "Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ...
7
votes
4answers
145 views

A new approach to find value of $x^2+\frac{1}{x^2}$

When I was teaching in college class ,I write this question on board . if we now $x+\frac{1}{x}=4$ show the value of $x^2+\frac{1}{x^2}=14$ Some student ask me for multi idea to show or prove that ...
0
votes
0answers
28 views

Sufficient conditions for finite-dimensionality?

Suppose we are given a subset $A$ of a complex vector space $\mathcal V$, and we are asked to look at the subspace $V$ (Hamel-) spanned by $A$. Of course, a spanning set may be very large compared to ...
9
votes
1answer
149 views

Counting number of mathematical objects and structures

Regarding the numbers of certain mathematical objects and structures, especially sets, relations and functions, I've compiled a list of the counts from various sources: Partitions of a set with $k$ ...
3
votes
2answers
49 views

Different proofs for the $L^2$ isometry of the Fourier transform on $\Bbb R$

Over the years I have come across several different proofs for the $L^2$ isometry of the Fourier transform (as it exists as an integral operator on $\Bbb R$). Often the traditional proofs hinge on the ...
2
votes
0answers
26 views

Analogies between finite groups and Lie groups

I believe there are some striking analogous facts between finite groups and Lie groups. One analogue almost too basic to mention is the appropriate notion of subobjects. In elementary group theory ...
5
votes
2answers
117 views

Applications of Hodge theory to topology and analysis

I am going to give a talk for the PhD students' seminar at my university. The audience is composed mainly by algebraic topologists, algebraic geometers and analysts. I have decided that I'm going to ...
6
votes
1answer
191 views

What does Hartshorne do wrong?

I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain ...
1
vote
0answers
15 views

Examples of generalized geometric series.

I'm preparing a short presentation on the convergence of the geometric series of matrices, and I'd love some examples of their uses. I've encountered them when approximating inverses of matrices ...
1
vote
2answers
65 views

Theorems relating to the limitation of mathematics

At one point, mathematicians believed that they may be capable of expressing all of mathematics in one system of ideas, and that their abilities were unlimited. Unfortunately, things like the Godel's ...
4
votes
1answer
130 views

Examples of bi-implications ($\Leftrightarrow$) where the $\Rightarrow$ direction is used in the proof of the $\Leftarrow$ direction.

[I'm asking for examples of proofs with a certain structure. There is quite a lot of text before arriving at the questions. This is because asking for examples of a phenomenon is best carried out by ...
14
votes
5answers
208 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 ...
59
votes
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 ...
3
votes
0answers
90 views

Exercises on group theory [closed]

What are some difficult, challenging and fair exercises in group theory? I know it is quite general, in particular I am referring to these areas of group theory: theory of automorphism group ...
10
votes
15answers
598 views

$2=1$ Paradoxes repository

I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one ...
2
votes
1answer
146 views

How many six digit numbers start with the same two digits and end with the same three digits?

Say that there is a 6 digit number the first digit is not allowed to be 0 or 1 so How many number combinations start with the same two digits and end with the same three digits ie.119333, 448222, ...
4
votes
6answers
206 views

What are some false proofs for true or false statements where the error in the proof is not obvious? [closed]

I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = ...
46
votes
19answers
6k views

What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are? Examples I can think of: Values of the Busy beaver function: It is a well-defined function, but not computable. It ...
15
votes
11answers
499 views

Different ways finding the derivative of $\sin$ and $\cos$.

I am looking for different ways of differentiating $\sin$ and $\cos$, especially when using the geometric definition, but ways that use other defintions are also welcome. Please include the ...
0
votes
1answer
50 views

Books that emphasize physical applications

I made a book recommendation thread recently which got deleted, so I'm hoping this one doesn't have the same fate as the question seems much more defined. I was reading an interesting article by VI ...
0
votes
0answers
24 views

Composition of non-differentiable functions to produce a differentiable one

What are examples of non-differentiable functions which when composed form a differentiable function? It would even better if they are $C^\infty$ I am hoping this will turn into a big list, if it ...
2
votes
1answer
47 views

Useful matrix inner products

What are some interesting/useful examples of matrix inner products on square complex matrices, other than $\langle A, B \rangle = Tr(A^{\dagger}B)$?
3
votes
1answer
64 views

List of textbooks that take a historical approach

As the title suggests my aim in this topic is to make a big list of textbooks on any mathematical topic that take a historical approach. I will start with the ones I know: Thomas Muir - The theory of ...
9
votes
0answers
142 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
4
votes
2answers
186 views

Applications of algebraic graph theory

What are some subtle, or non-obvious applications of algebraic graph theory? Obviously it can be used to study anything directly involving graphs (for instance, the wikipedia page mentions ...
1
vote
0answers
66 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
9
votes
5answers
138 views

$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity \begin{align*} ...
8
votes
3answers
357 views

Elementary topology examples

I'm preparing (to teach) my first class of undergraduate topology and I'm looking for some elementary, motivating applications of topology for the first day. We'll be following Munkres, starting with ...
5
votes
1answer
147 views

A beautiful book on arithmetic doesn't treat you like a little baby

The state of arithmetic today is disgusting. The textbooks on it are absolutely repelling, the authors treat it like a subject that will be of concern to only babies. They don't show any love, they ...
3
votes
7answers
303 views

Alternate ways to prove that $4$ divides $5^n-1$

I was working for various method to solve this: For all $n\in \mathbb N$: $4\;\mid\;(5^{n}-1)$. My try was: 1st: $$n=1 \to 4|5^1-1\\n \geq 2 \to 5^n=25,125,625,3125,...\\ n\geq 2 \to ...
1
vote
2answers
96 views

What are some unfamiliar and/or special tricks used to evaluate limits?

What are some neat tricks used to evaluate limits that might be otherwise a problem to deal with? I'm not asking for methods akin to L'Hopital's rule.. which is often used. My question is geared ...
0
votes
4answers
177 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
2
votes
1answer
166 views

(Theoretical) Complex Analysis Textbooks [closed]

Most books I've seen on complex analysis do not develop it theoretically, which can be somewhat infuriating for the budding pure mathematician. What I am looking for are some comprehensive, rigorous ...
23
votes
15answers
1k views

New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$

Given $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$. 1st Proof: Let $s$ be defined as $$ s=1+2x+3x^2+4x^3+5x^4+\cdots $$ Then we have $$ \begin{align} ...
0
votes
0answers
218 views

Final year dissertation/project ideas for numerical methods

In my final year, I have to submit a project/dissertation on Numerical Methods. I have done a course on it, which included some proofs and programming. Just eager to get ideas that I can look at. PS ...
4
votes
2answers
631 views

Interesting facts and problems to motivate high school combinatorics students

I will give some classes in combinatorics to high school students and I would like to know some facts (and proof) I can show to my students to motivate them to study this beautiful subject. I'm ...