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5
votes
4answers
420 views

Intuitive Explanation of Morphism Theorem

Is there an intuitive explanation for the morphism theorem from introductory abstract algebra? First Morphism Theorem: Let $K$ be the kernel of the group morphism $f: G \to H$. Then $G/K$ is ...
5
votes
2answers
303 views

The complement of a torus is a torus.

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can ...
4
votes
3answers
364 views

Interesting theorems/facts about identification spaces

I am now studying algebraic topology (still at the beginning). I am now studying identification spaces, adjunction spaces,... As I still don't know how these concepts are going to be used, I think I ...
6
votes
8answers
2k views

Final year project ideas - complex analysis

For my final year, I have to do a project for a module. I want to investigate something in the complex analysis area. I've only covered the basics of analysis, like Cauchy's IT/IF, residue theorem ...
3
votes
1answer
87 views

Expected Value of Students on a Bus

There's a question in my probability book that says there are $148$ students on $4$ buses containing $40, 33, 25, 50$ students, respectively. If we let $X$ denote the number of students that were on ...
3
votes
2answers
91 views

Pre-test preparation [closed]

I'm currently an undergrad in maths and I wanted to know how those who have completed their degrees handled their test preparation. That is, how did you study and how did you manage the anxiety if ...
27
votes
4answers
18k views

Pure Mathematics Vs Applied Mathematics Job Prospects

I need and request some advice on selecting the area for my research. In particular, I want to know the about job prospects of Applied Mathematics Versus Pure Mathematics in Academia. I understand ...
3
votes
1answer
217 views

Providing a sketch for a proof before proceeding through the actual proof. [closed]

Question is pretty straightforward. My mathematics is sloppy, and I recognize my inaptitude in that my proofs are more or less too intuitive. My diagnosis dictates the fact that I attack a problem ...
1
vote
1answer
120 views

Does π depends on the norm? [duplicate]

If we take the definition of π in the form: π is the ratio of a circle's circumference to its diameter. There implicitly assumed that the norm is Euclidian: \begin{equation} ...
2
votes
1answer
257 views

Is there an intuitive non-mathematical way to show $\sum_n 1/n=\infty$?

I want to show some school students about the sum of the harmonic series diverging and I need some nice interpretation for this. I'll preferably love to have a figure. Thanks in advance!
8
votes
3answers
495 views

Math under time constraints [closed]

I'm currently an undergrad math student and I have been wondering if being able to perform well under time constraints requires a deeper knowledge of the material than performing under no time ...
17
votes
5answers
1k views

Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ...
2
votes
3answers
143 views

teaching concept of “subspace” to linear algebra students

I'm currently a T.A. for an introductory combined linear algebra & differential equations course geared toward engineering students. One major problem I've run across is that most of my students, ...
10
votes
1answer
233 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
3
votes
1answer
86 views

When asked to find all solutions to a differential equation, what's left implicit?

Suppose the following problem were to appear in an introductory textbook on differential equations. Find all solutions to $y'=y$. What's left implicit in such a statement? My interpretation of ...
8
votes
2answers
312 views

Currently, what is the largest publicly known prime number such that all prime numbers less than it are known?

So recently, an absurdly large prime number was found, but a lot of prime numbers less than it are still not known. I am wondering up to where we know all the primes. I put "currently publicly known" ...
10
votes
1answer
671 views

How long does it take to write a math paper? [closed]

I am currently working on an applied mathematics paper (with my supervisor, I am a student) that is long overdue: supposed to take 4-5 months but I have been working on it for 6.5 months. I was ...
8
votes
1answer
147 views

Proof vs Practice

I've been brushing up on a lot of basic arithmetic, algebra, and logic as I work towards a review of calculus (and beyond), and I keep noticing that in order to fully understand many principles in the ...
2
votes
5answers
453 views

Geology with maths [closed]

Can anyone suggest me topics that connect maths with geology or geography or anything related to earth? Thank you. ...
20
votes
1answer
381 views

Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
12
votes
3answers
278 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
6
votes
1answer
216 views

How to explain that $\Bbb{R}$ is not countable to a non-mathematician

What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians? I ask this because these days I'm in a debate with someone ...
7
votes
1answer
559 views

Problem regarding mathematical skill of a math lover [closed]

I don’t know whether it would be a good idea to seek solution for some personal problem here. But as there’s no harm in asking I’m putting it under the soft tag. It’s about forming a career in ...
2
votes
1answer
294 views

Selecting Differential Geometry Exercises

I'm self-studying differential geometry with Do Carmo's books "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry" and I find those books very good, however I feel a little ...
28
votes
3answers
868 views

Do free online collaborative solution manuals exist?

I'm not a mathematician by training and a rarely come in contact with mathematicians. For this reason I find solution manuals to be incredibly useful - reading them allows me to see how experienced ...
1
vote
1answer
93 views

“anti-Cartesian” method in plane geometry

I have got very soft question. It's true that all plane geometry problems had analytical (easy) solutions. And everyone can take Cartesian coordinate plane and count all problems for $<\infty$ ...
1
vote
1answer
203 views

Correct way of saying that some value depends on another value x only by a function of x

I would like to know what good and valid ways there are to say (in words) that some value f(x), which depends on a variable x, in fact only depends on x "through" some function of x. Example: For ...
2
votes
1answer
31 views

Is every GO-space collectionwise normal?

Is every GO-space collectionwise normal? And what is the relation between collectionwise normal and monotonically normal? I know a GO-space is always monotonically normal. Thanks.
2
votes
2answers
935 views

For homework questions what is the difference between being asked to verify something and being asked to prove something?

I've always been curious if there is a definite difference between the terms or if they just depend on the context of a problem.
12
votes
4answers
1k views

Fermat's 'proof' of his Last Theorem

The definition of a unique factorisation domain came up in my rings lecture about a week ago, and my lecturer mentioned that Fermat's 'proof' of his Last Theorem probably relied on the (false) ...
10
votes
4answers
2k views

Should I try to change the way Abstract Algebra is taught at my university? If so, how?

[This (soft) question should be Community Wiki.] Background: A year ago, I did a one-semester long course on Abstract Algebra at my university. When we started, I was excited, because I knew the ...
2
votes
2answers
135 views

What should I know about Sobolev space?

I have done some Sobolev spaces with some embedding theorems, trace theorems etc. Sorry that my question is really vague. If my professor asks me what is great about Sobolev space, what should I ...
5
votes
4answers
2k views

Sequence Notation — Which brackets to use?

I'm teaching sequences at the moment. I've always put sequences in round brackets, for example $(1,2,3,4,5)$ is a sequence whose first member is $1$, whose second member is $2$, and so on. I've also ...
3
votes
1answer
336 views

Paris VI or ENS Lyon for master's student interested in algebra?

I have the opportunity to study in France during the forthcoming autumn, either at Paris VI (6), or at ENS Lyon, and I have trouble deciding which offer I should take. I'm mainly interested in algebra ...
16
votes
4answers
492 views

Mathematical Habit

Whenever I have to do proofs or understand a concept, I start with examples and generalize. As an aspiring mathematician, I have fears that it might hurt me, especially when we have so many ...
595
votes
158answers
37k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
15
votes
0answers
243 views

Writing Hard Problems [duplicate]

Having done a fair bit of contest- type math, I was interested in trying to write similar sorts of problems, only to find myself immediately stumped. How does one write "elegant problems" of the kind ...
5
votes
0answers
871 views

Publication date for Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
2
votes
2answers
1k views

What are the drawbacks of multiple-choice questions? [closed]

I can easily understand the advantage of multiple-choice questions for instance in grading and so. A drawback is that real life problem don't have multiple choice questions all the time for instance ...
6
votes
0answers
213 views

The high road to learn algebraic geometry

Suppose that a student has a basic knowledge in commmutative algebra at the level of the Atiyah-MacDonald. What do you think about the following steps to learn algebraic geometry? step 1: read ...
9
votes
8answers
318 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
26
votes
3answers
842 views

Intuition on fundamental theorem of arithmetic

I'm sorry ahead if time if this is overly trivial for this site. Currently in school, much of what I enjoy is number theory - based. Currently, I lean pretty heavily on the FTA for a good deal of my ...
11
votes
0answers
417 views

Approximating a blackboard with long-distance communication

From time to time I need to talk about mathematics with my advisor remotely. I would like to approximate writing on a blackboard together as closely as is reasonable. What are some technological ...
3
votes
2answers
185 views

Which set of subjects has the most algebraic 'flavour' to it?

I'm finishing my undergraduate mathematics programme this summer. The graduate program at my university requires that you specify a specialization on entry. This is very difficult for me and would ...
5
votes
4answers
395 views

What pattern does set theory study?

If mathematics is the science of patterns, what pattern does set theory study? My thoughts so far: set theory studies the pattern of relationships between members of a collection and between ...
0
votes
2answers
64 views

If we can imagine $\mathbb{C}$ as a 2-dimensional Euclidean Space, then can we imagine $\mathbb{C}^2$?

It is easy to think of $\mathbb{C}^2$ as an ordered pair. I just wonder if it is possible to put $\mathbb{C}^2$ into illustration, since $\mathbb{C}$ has taken the role of two dimensional Euclidean ...
0
votes
1answer
43 views

Understanding the relation between countably paracompact and monotonically normal

Does monotonically normal imply countably paracompact? Thanks ahead:)
7
votes
1answer
113 views

The collection of pathological examples in one reference - Reference request

I would like to ask whether there is a reference which collects pathological examples in mathematics (in general). What I mean is that, for instance, consider Weierstrass function. It has the ...
3
votes
2answers
174 views

How is “1” defined in various branches of mathematics?

Wikipedia does not elaborate much on the concept of "One" in such branches as graph theory, ring theory, algebra, topology, measure theory, formal logic, etcetera. How can one grasp the concept of ...
3
votes
2answers
117 views

Conditional events that are not in the event algebra?

The Wikip. page on conditional event algebra states that: David Lewis showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any ...