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24
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
17
votes
21answers
1k views

Concepts in mathematics which are referred to as 'generalizations' [closed]

I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late ...
505
votes
46answers
312k 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 ...
38
votes
12answers
16k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
9
votes
2answers
101 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
-4
votes
0answers
44 views

Why can't infinity be a number? [duplicate]

Why can't infinity be a number? All reasons why it can't that I come up with, require that it already isn't one. Yes you are right, by definition $\infty$ is larger than all numbers, take $\infty ...
0
votes
1answer
43 views

Universe in a single equation [closed]

I have got a very abstract question. We all see the world around us, it's made up of various shapes, curves, et cetera. My question is, can every shape, self-intersecting or not, be represented by a ...
32
votes
19answers
2k views

Which mathematicians have influenced you the most?

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, ...
7
votes
3answers
296 views

Number Theory Reading List

What are the essential number theory texts that every serious student of number theory should read?
3
votes
1answer
106 views

Is there a system of mathematics where everything is a function?

I was wondering if there is a system of mathematics where everything (except sets) is a function. For example, 3 would be the 3 function $x \mapsto 3$. There would be basic operators, such as $+$, ...
47
votes
10answers
5k views

Why are primes considered to be the “building blocks” of the integers?

I watched the video of Terence Tao on Stephen Colbert the other day (here), and he stated, like many mathematicians do, that the primes are the building blocks of the integers. I've always had ...
1
vote
1answer
41 views

Why is “random” in the definition of discrete random variable?

We defined discrete random variable as follows: Suppose $S$ is a countable sample space. Then a function $X:S\to R$ is called a discrete random variable. The lecturer made a note that the "random" ...
0
votes
2answers
62 views

How to brush up on calculus?

It's been years since I took calculus, and while I have a good understanding of the theorems of single variable calculus from my real analysis courses, computationally I am a bit slow. It takes me ...
0
votes
2answers
34 views

Solve $x^3-ax=by$ If $\gcd(x,y)=1$

Solve the diophantine $x^3-ax=by$ If $\gcd(x,y)=1$. Any hint? My first impression is $\gcd(x,b)>1$ and $x^2=ky+a$ for some integer $m$. I conclude that as long as there exists a square integer ...
1
vote
0answers
33 views

Can I apply measure theory in non-mathematics fields?

I am working in a field where researches try to get insight about a complex process. I will give an example to demonstrate this. Let's say, we are attempting to get the most efficient and cost ...
0
votes
0answers
25 views

What other words are good replacements for 'heat' in the phrase “heat equation” (the famous PDE), apart from 'diffusion'?

Historically, the term 'heat' has value as it hearkens back to the context in which Fourier studied the heat equation. Pedagogically, the term 'diffusion' has value as it imparts a more general, ...
24
votes
6answers
561 views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
8
votes
3answers
151 views

Breaking symmetries

Back when I was studying electromagnetism and Maxwell's equations, our teacher told us a quote. I can't recall it exactly, but the meaning was roughly the following: Symmetry in a problem is ...
4
votes
0answers
45 views

Algorithm for finding “fact families”

My friend's 3rd grader encountered the following question regarding "fact families" on her math homework: I was in 3rd grade sometime in the 1980s, so I don't believe I ever encountered this term ...
6
votes
1answer
392 views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
2
votes
1answer
89 views

Interesting Olympiad Questions.

Rather than through research, I much prefer discovering new maths or interesting theories through doing problems and I also enjoy contest maths which has led me to this question: Which (high school) ...
6
votes
0answers
58 views

How hard is it to succeed in pure math research after an adviser's death midway through PhD? [migrated]

TL;DR: How hard would it be to finish out a Ph.D. and have an effective early career as a pure mathematician after my adviser has unexpectedly passed away? My department does not have anyone else able ...
48
votes
13answers
3k views

How to stop forgetting proofs - for a first course in Real Analysis?

I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read ...
3
votes
2answers
137 views

Extremely hard and stimulating (undergraduate) real analysis $problems$

To put it simply: I have heard of many problem books in real analysis (also on this website), but the exercises they propose seem quite standardized. What are problem books that propose really ...
1
vote
0answers
53 views

How should one ask for first names of professors? [migrated]

This is probably a soft question. I am interested in complex analysis, and I want to ask the first name of this person, S. Ponnusamy, whose name I wish to mention in a paper. What is the best way to ...
2
votes
0answers
27 views

Are there any applications for complex analysis in population dynamics?

Strange question: could complex analysis be used to understand population dynamics? I'm interested in modelling dominance hierarchies, mating relationships, and illness behaviour in ancient ...
0
votes
2answers
37 views

Problem Solving Practice

Are there any resources ( mainly books, but other things as well :)) which involve a lot of exercises/problems. These problems should be challenging but still solveable for a High School Student. The ...
0
votes
0answers
34 views

Another look at the trinomial of the form: $ax^n+bx+c=0$

Has the trinomial of the form $ax^n+bx+c=0$ been fully studied for $n>2$? If so, please let me know of any reference or interesting findings. Thanks.
4
votes
2answers
181 views

Dynamical systems and differential equations reviews/surveys?

I would be very glad if someone could point me to modern reviews/surveys on these topics. To be concrete, I'll provide some examples: S. Smale, Differentiable dynamical systems D. V. Anosov, On the ...
0
votes
1answer
308 views

Does this paper have any mathematical value? [closed]

Apparently every so often someone posts on MathOverflow questions involving some impossible to understand mathematical formulas, and always referencing to this paper (I am unable to link any of the ...
0
votes
1answer
49 views

How to get better at intermediate and difficult pencil-and-paper calculations?

Out of pure curiosity I recently had a look at some old Cambridge mathematical tripos questions from the 19th century. Those guys certainly knew how to handle difficult algebraic (and even arithmetic) ...
3
votes
4answers
96 views

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$?

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$? I was trying my best to do the problem but like I don't know where to start or anything!
79
votes
6answers
4k views

Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
0
votes
0answers
45 views

Apostol's Calculus Vol II OR Hubburd's multivariable OR Shifrin's multivariable for self study

I'm trying to self study multivariable calculus which I took at university but mostly forgot about it! I'm looking for a textbook that also incorporates linear algebra and gives a coherent view of the ...
3
votes
1answer
73 views

Applications of computer science to mathematics

I have been introduced to algorithms, computability and computational complexity (as part of my minor in CS). What are some mathematical topics that I can tackle with the new perspectives I ...
1
vote
1answer
70 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
0answers
45 views

Unexpected applications of row rank = column rank

The fact that the row rank of a matrix is the same as the column rank is quite surprising (to me atleast, hopefully to you too!). I am looking for unexpected applications of this fundamental fact, ...
0
votes
0answers
18 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
27
votes
7answers
3k views

Chatting about mathematics (with real-time LaTeX rendering)

Do you know about some tools which can be used for online chat about mathematics? In particular, I am interested in software which would be able to render LaTeX formulas. (Since LaTeX is probably the ...
1
vote
2answers
85 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
702
votes
28answers
47k views

Can I use my powers for good?

I hesitate to ask this question, but I read a lot of the career advice from mathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that ...
3
votes
1answer
26 views

How can I improve my explanation that the ratio $\theta=\frac{s}{r}$ that defines the radian measure holds for all circles?

I'm trying to demonstrate why the ratio $\theta=\frac{s}{r}$ (where $s$ is an arc measuring some $s$-units in length and $r$ is the radius of the circle) which defines the radian measure holds for all ...
6
votes
7answers
612 views

A book for abstract algebra with high school level

Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on ...
1
vote
1answer
62 views

A more advanced book on real analysis

I have already studied "Advanced Calculus Second Edition by Patrick M. Fitzpatrick" on real analysis. It is the best book on real analysis I found that can be studied by self-learning with high-school ...
1
vote
2answers
155 views

How can I improve my explanation of why the ratio $\pi=\frac{C}{d}$ holds for all circles?

I'm trying to informally explain why $\pi$ holds for all circles. I would like to know if there is anything pertinent that I can add, or that is wrong with this explanation. It's an explanation, not ...
11
votes
6answers
462 views

Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like ...
3
votes
1answer
84 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
7
votes
2answers
850 views

The way into set theory

Given that I am going through Munkres's book on topology , I had to give a glance at the topics included in the first chapter like that of Axiom of choice, The maximum principle, the equivalence of ...
6
votes
3answers
329 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

I'm searching for some material (books or lecture notes) that extensively uses a geometric approach to explain the meaning of the concepts realted regarding to vector spaces, matrices, and linear ...
2
votes
2answers
175 views

Little, unknown, English or French research journals with good mathematics

In this article by Gian-Carlo Rota, you can read: "I bought a copy of Frederick Riesz' Collected Papers as soon as the big thick heavy oversize volume was published. [...] It was clear that ...