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0
votes
2answers
31 views

When is the commutator subgroup a maximal subgroup?

Let $G$ be a group , when can we say that the commutator subgroup $[G,G]$ is a maximal subgroup ?
13
votes
2answers
465 views

How much time does a great mathematician take to solve an extreme problem?

I really love math, and I can spend hours, days or even years to solve a really simple problem if I can't do it. However, there are certain problems, which I am not able to solve in an hour or so. It ...
1
vote
0answers
68 views

Which book would you recommended as help (assistance) for reading the so-called “Tohoku Paper”?

Recently I thought that maybe is a good time to try, read Grothendieck's "Tohoku paper" as a sort of inspiration for the future and to read some of the ideas of this great mathematician, which (among ...
5
votes
2answers
148 views
+50

Future learning for a math graduate in applied mathematics references

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications ...
31
votes
6answers
5k views

What are some interpretations of Von Neumann's quote?

John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. ...
1
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0answers
40 views

In mathematics, what is a placeholder?

Google defines the word placeholder in the following image: . My question is: Is this the only definition of the word placeholder in mathematics? I am thinking along the lines: If $M = p^k m^2$ ...
6
votes
1answer
92 views
+100

Preserving equality between different mathematical objects.

I'm taking an 'Intro to Higher Mathematics'-type course right now, were we learn about basic set theory, number theory, algebra, etc. and I had the following thought: Say you're trying to solve a ...
2
votes
2answers
66 views

How could I show that if $a$ is an integer, then $a^3 \equiv 1, 0, 6 \mod 7$? [duplicate]

I've literally tried everything including proofs by cases. If I were to try a an odd integer, then I would get $8k^3 + 4k^2 + 8k^2 + 4k + 1$, which obviously couldn't convert to a $\mod 7$, and I'm ...
105
votes
33answers
13k views

What are the most overpowered theorems in mathematics? [closed]

What are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I'...
-4
votes
0answers
26 views

Is there any short quote on influence of computer in algebra? [on hold]

I need a short quote on algebra (or a short quote on influence of computer in algebra).
1
vote
1answer
61 views

Conceptually tough calc 1 problems; where are they hiding? [on hold]

I'm finding myself falling into the trap of memorizing some sort of cookbook recipes in calculus class instead of using my brain. I'd really like to find some conceptually challenging calculus 1 (...
4
votes
1answer
68 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
1
vote
1answer
14 views

Books with problems on extremal principle

Now, I've seen and read quite a few problem solving books but Arthur Engel's 'Problem Solving Strategies' is the only one I've seen where the extremal principle is treated. (Unlike the pigeonhole ...
2
votes
0answers
46 views

Simple proof of Newton identities

The functions $s_1=x_1+x_2+\cdots +x_n$, $s_2=\sum_{i<j} x_ix_j$, $\cdots$, $s_n=x_1x_2\cdots x_n$ are elementary symmetric functions in $x_1,x_2,\cdots,x_n$ (or more precisely, elementary ...
0
votes
1answer
30 views

Why is the condition (L) is called condition (L)? [on hold]

For a directed graph, if every cycle has an exit, the graph is said to be satisfying condition (L). Why is condition (L) is called condition(L)? What is the meaning of (L)?
43
votes
11answers
3k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
52
votes
11answers
3k views

Is it not effective to learn math top-down?

By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving ...
2
votes
3answers
166 views

What are the most obscure or advanced mathematics with practical application

Throughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no ...
18
votes
5answers
1k views

How to improve accuracy when solving calculus questions

I find calculus to be a really interesting topic to study, and from what I've experienced it simply boils down to applying algebra to more complicated concepts. I understand calculus and can easily ...
2
votes
0answers
46 views

Online Video Lectures for Graduate Level Mathematics?

I am wondering if there is a nice compilation of good video lectures in graduate level mathematics? I mean a website, or a forum, or maybe something like course-era (which is mostly undergraduate ...
2
votes
2answers
314 views

Why are quadrants defined the way they are?

I was thinking about planes and things, and suddenly wondered why quadrants are defined the way they are, the first on the top-right, and so on. I wonder if this gives us any benefit, or if any ...
2
votes
1answer
50 views

What is the intutition behind the negative exponential ? in linear logic?

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...
1
vote
0answers
33 views

How to find a simple function with those very specific properties?

I'm looking for a function F : N -> N, for N < 10, such that: ...
3
votes
1answer
94 views

Are there any modern mathematicians whose research interest is in “Probability Theory”?

I have seen professors in universities list "stochastic calculus", "stochastic analysis", "stochastic processes", "stochastic geometry" and "applied probability" as research interests, but are there ...
0
votes
1answer
403 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
2
votes
0answers
32 views
+50

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
2
votes
1answer
76 views

Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis? For instance, every example I can think of in analysis is first-countable. I don't think it can be metric ...
5
votes
6answers
627 views

How to Axiomize the Notion of “Continuous Space”?

EDIT (to clear up controversy and misunderstandings caused by my poor wording): Historically, Riesz's efforts to try and make rigorous a notion of a "continuous space" (as opposed to "discrete ones") ...
1
vote
3answers
221 views

Mathematical doodle games

Vi Hart's doodling videos and a 4 year old son interested in mazes has made me wonder: What are some interesting mathematical "doodling" diversions/games that satisfy the following criteria: 1) They ...
3
votes
1answer
192 views

Textbook +reference book in complex analysis

Which book can be used as an introductory textbook in complex analysis? I have some choices (more suggestions are welcomed) Marsden & Hoffman J.B. Conway Ahlfors Palka Lang Stein & ...
17
votes
11answers
1k views

Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
1
vote
2answers
45 views

Is this simple looking complex expression valid always?

$$ z^{a+ib} = z^a*z^{ib} \hspace{2mm} \forall z\in \mathbb{C} $$ In high school I was always taught to see the + in complex numbers as analogous to that is reals. However can it be proven to be ...
4
votes
0answers
43 views

Is there a schsim between classical logic and categorical logic?

I've been trying to learn a little bit more about the foundations of mathematics, and it has strike me that there seems to be two competing points of view about what the foundations should be. While ...
11
votes
4answers
814 views

Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math? [on hold]

So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion. For example, I have no idea what eigenvalues, ...
0
votes
1answer
75 views

Weak version of Cauchy's theorem in complex analysis

Let $f(z)$ be a complex valued function defined on an open subset $U$ of the complex plane. We say $f(z)$ is analytic if it can be expressed by a convergent power series on a neighborhood of every ...
42
votes
12answers
4k views

Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
1
vote
0answers
39 views

Is it wise to select a research topic first and then choose an advisor? [on hold]

I am a student of Pure Mathematics. I am interested in Linear Algebra and Abstract Algebra. As I was browsing the Internet for some good research topics which contains a flavor of linear algebra ...
5
votes
1answer
268 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
52
votes
10answers
5k views

Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
0
votes
0answers
13 views

Advice for Self-Study(application in financial engineering) [on hold]

I am currently studying statistics and I have such background: 1) Single and multivariable calculus (Stewart Calculus book) 2) Linear algebra(Strang's textbook) 3) Theory of probability(Ross book) ...
3
votes
1answer
52 views

Is there a searchable database of mathematical objects that you can search by property?

For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function....
5
votes
0answers
98 views
+50

Algebraic Geometry Project ideas related to Computer Science

I am a Computer Science Undergrad student with an interest towards Algebraic Geometry.I have just recently started and am currently reading Miles Reids' Undergraduate Algebraic Geometry(I have read ...
7
votes
3answers
102 views

Why do we study the number of homomorphisms/isomorphisms between fields?

From the first abstract algebra class, we encounter many problems that ask us to find the number of homomorphisms/isomorphism a between two algebra structures (e.g. field). My first question is why ...
2
votes
0answers
53 views

Future Space Opportunities for a Mathematician [on hold]

I don't know if this question should be asked here or on "Mathematics Educators", however I'll post it here for the moment. I've just finished my first year of Mathematics and I do really like maths. ...
96
votes
20answers
20k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
2
votes
1answer
91 views

How to organize myself around calculus?

Calculus is the most advanced topic I have encountered in math. The book that I am using is clear as can be, but it has so many definitions and theorems. I would like to have all the most crucial ...
1
vote
1answer
64 views

Disorganization In Mathematics

I was just wondering why it is that there are so many overlapping and seemingly random terms in mathematics. For example, I'm learning graph theory and according to different notes or books, two edges ...
31
votes
4answers
2k views

Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.

In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\...
11
votes
1answer
40k views

Why is a full circle 360° degrees? [duplicate]

What's the reason we agreed to setting the number of degrees of a full circle to 360? Does that make any more sense than 100, 1000 or any other number? Is there any logic involved in that particular ...