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5
votes
1answer
713 views

When addition distributes over multiplication

Everyone is familiar with distributivity of multiplication over addition of real numbers. The distributivity of two binary operations sometimes goes both ways (e.g. max and min, or for lattices in ...
1
vote
2answers
4k views

Understanding $\epsilon$ transitions in a finite state automaton

I am trying to understand how $\epsilon$ transitions work. From what I've read, when you "go" to a state S that has arrows pointing outwards with $\epsilon$'s in it, you automatically go to those ...
8
votes
2answers
496 views

Pull the teeth out of Lebesgue integration

In Lebesgue integration we usually approximate the function we want to integrate with step-functions on measurable sets. How much "power" do we take away if we require that the step functions are on ...
45
votes
14answers
9k views

How to effectively and efficiently learn mathematics

How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours? (Apologies in advance if the question is ill-posed or too subjective in its current ...
6
votes
2answers
358 views

Is it adequate to expect to accomplish some amount of original work in a masters thesis? [closed]

Is it adequate to expect to accomplish some amount of original work in a masters thesis?
74
votes
8answers
20k views

Learning mathematics as if an absolute beginner?

I dread mathematics, and I believe it's because I have come to associate mathematics with the experience of terrible teachers. All of my math teachers have been grumpy, but one in particular was the ...
4
votes
2answers
245 views

Can there be such a thing as a classification of classification theorems?

Can there be such a thing as a classification of classification theorems?
5
votes
1answer
399 views

Reading commutative diagrams?

Sorry for this whole bunch of questions. Please note, that I know what a commutative diagram is, and that I can somehow read them, at least the simpler ones. But often enough the diagrams are labelled ...
2
votes
1answer
81 views

a function of a dependent type, a section, a sheaf

I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...
29
votes
5answers
3k views

How to pick a thesis advisor?

This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due ...
11
votes
6answers
7k views

In what order should the following areas of mathematics be learned?

I am in a biological field (medicine) but I have genuine passion for mathematics. I want to learn it on my own , in my spare time. Mathematics , as I gather, is learned best when you have grasped ...
17
votes
3answers
747 views

General question about mathematical thinking

I am starting to see a pattern in my math studies, and I'm interested in your thoughts on the subject. It seems like there are certain things that are much easier for me to learn than others. For ...
62
votes
6answers
3k views

How Do You Actually Do Your Mathematics?

Better yet, what I'm asking is how do you actually write your mathematics? I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up ...
14
votes
8answers
1k views

Elementary Papers at ArXiv

Inspired by this question, at MO i am asking this question. Can anyone list some elementary articles at ArXiv which can be understood by High-School/Undergrad Students. I am asking this because, i ...
6
votes
1answer
728 views

two phds in math

If one gets a phd in math, is it able to be admitted to another phd program in math? It is not for me, for someone else.
4
votes
2answers
236 views

Intuitionism and circles

I know next to nothing about intuitionism, so my question is probably silly :) As I understand from Wikipedia, intuitionism (at least finitism) doesn't 'trust' in the existence of irrational numbers, ...
5
votes
2answers
340 views

about career choice

What possible choices do fresh Phds have if they can't find a job,especially for pure math,provided current job market is tight?
3
votes
3answers
179 views

Motivating differentiable manifolds

I'm reading lectures for the first time starting next week ^_^ The subject is calculus on manifolds I was told that the students I'll be lecturing are not motivated (at all), so I need to kick off the ...
45
votes
3answers
4k views

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
10
votes
4answers
2k views

Why is an integral domain a commutative ring with unity?

I was wondering why an integral domain is required to be commutative and have a multiplicative neutral element. For example the quaternions would be a non-commutative integral domain. The ...
8
votes
3answers
479 views

Explanation of a Phrase from Prof. Ravi Vakil's Website

I was just browsing through Ravi Vakil's website when i found a nice article written on what he demands from his students. Here is the webpage. (For those interested!) ...
13
votes
3answers
968 views

Guidelines for learning about Ramanujan's work?

It is well known that one of the first books Ramanujan studied was "Synopsis of Pure and Applied Mathematics" and that it shaped the way Ramanujan thought and wrote about mathematics. Being interested ...
5
votes
10answers
4k views

Does a negative number really exist?

Second Update: I see that some answers that reference my image are more closely answering my question. Here is a second image to clarify my point. Take this image representing a checkerboard like ...
11
votes
9answers
962 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 ...
5
votes
2answers
404 views

On the Möbius $\mu$ function

A search on wikipedia shows: $$\mu(n) = \sum_{k=1,gcd(k,n)=1}^{n} e^{2\pi i \frac{k}{n}}$$ But that uses complex numbers... and requires finding out the gcd... How useful will be a method, if that ...
27
votes
10answers
2k views

Contributions of Galois Theory to Mathematics

What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories. ...
5
votes
1answer
256 views

Are there other pseudo-random distributions like the prime-numbers?

Does there exists other structures in math, which are seemingly random, but deterministic, and follow rules similar to the prime numbers, by rules I mean there must be statements similar to goldbach's ...
29
votes
8answers
3k views

Why should we care about groups at all?

Someone asked me today, "Why we should care about groups at all?" I realized that I have absolutely no idea how to respond. One way to treat this might be to reduce "why should we care about groups" ...
2
votes
4answers
228 views

Choosing subjects at high school

For a high school student that has the options of taking one of Statistics, Algebra or Calculus as subjects, which subject would be the best choice if the person is still undecided on a specific ...
136
votes
28answers
19k views

List of Interesting Math Videos/ Documentaries

This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list. Story of ...
3
votes
2answers
319 views

For teaching: Combinatorial Construction Riddles

Can you give examples of combinatorial construction riddles, approachable by gifted high school students? Examples: Find a finite set $A$ and $B \subset 2^A$ such that any element of $A$ is covered ...
3
votes
1answer
505 views

How to check the veracity of arXiv.org papers?

Suppose I found a preprint on arXiv.org (like this one: http://arxiv.org/abs/math/0309146 ), and I need to check if what's in there is actually true. Any tips? I don't want to study an article only to ...
7
votes
3answers
488 views

Why are they called “Isothermal” Coordinates?

If I understand correctly, Gauss proved that given any oriented Riemannian surface, one can find a complex structure on the surface so that the metric on the charts is just $f|dz|$, where $f>0$. ...
19
votes
3answers
702 views

Why isn't the gamma function defined so that $\Gamma(n) = n! $?

As a physics student, I have occasionally run across the gamma function $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$ when we want to generalize the concept of a factorial. ...
5
votes
2answers
365 views

Map of Mathematical Logic

My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a ...
8
votes
4answers
610 views

What material is good for extra-studying

I am an undergraduate Math student, i find the courses that I am in are too easy (calculus, linear algebra, Discrete math, logic). I want to become a successful mathematician. I want to know what ...
33
votes
17answers
3k views

Non-Scientific questions solved by mathematics

I have a general question about the applications of mathematics. What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political. I am ...
3
votes
1answer
1k views

Math for “Electrical Engineering & Communication Technology” recommendation

I have the feeling that an engineer should be more comfortable with math, than I'm felling I am right now. I also have a tendency toward learning math not required by the math classes I took, and was ...
2
votes
2answers
269 views

Where can I get the OLD (1940-1970) high school scholarship question papers?

Today matematics are moving away from understanding basics & core methodology. Level of the difficulties are reducing. Questions are becoming more simpler. Hence I would like to see the difficulty ...
10
votes
2answers
543 views

Category Theory with and without Objects

Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an ...
3
votes
0answers
77 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
10
votes
3answers
892 views

What is the key to success for a mathematician?

What is the most recommended quality a mathematician should have? Extremely high IQ levels?Passion for what he does?Patience and "stuborness"?Something else? Of course all of them are necessary ,but ...
10
votes
7answers
3k views

Topic for a high school-level math elective?

I'm looking for ideas for a 15-hour mathematical enrichment course in a Chinese high school. What (fairly) elementary subject would you suggest as a topic for such a course? ...
11
votes
3answers
1k views

Nice application of the Cauchy?-Frobenius?-Burnside?-Pólya? formula

Burnside's Lemma, whose list of names is longer than the proof, says that the number of orbits of a permutation group is the average number of fixed points of its elements. It's a very elegant result, ...
59
votes
7answers
5k views

Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?

As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem ...
14
votes
2answers
2k views

Summer programs to gain research experience as an undergraduate, outside the United States?

There are many summer programs in the United States, targeted at good, motivated undergraduate students majoring in mathematics. Most programs, however (if not all), demand that the applicant is ...
0
votes
2answers
106 views

how can I present an idea about the difference between two functions clearly?

I have a presentation in which I want to point out a difference between two functions. Instead of putting the two functions on a slide and pointing to the differences, I want to do something simple. ...
43
votes
8answers
3k views

Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock." There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational. ...
6
votes
2answers
296 views

which job to find to increase teaching experience

If you don't have enough teaching experience, what kind of job you should try to find to increase the teaching experience. Now many jobs have set requirements for teaching experience. It seems ...
14
votes
6answers
3k views

Books to learn physics, being a math major

What books would you recommend to learn physics, being a a Math major, from classical mechanics, electricity, etc. to modern physics?