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8
votes
5answers
961 views

Why do mathematicians use this symbol $\mathbb R$ to represent the real numbers?

So, I'm wondering why mathematicians use the symbols like $\mathbb R$, $\mathbb Z$, etc... to represent the real and integers number for instance. I thought that's because these sets are a kind of ...
4
votes
3answers
2k views

Why is it that I cannot imagine a tesseract?

I try hard to "visualise" (say "imagine") a tesseract but I can't. Why is it that I can't? This may be a question for a scholar of some other discipline and not for a mathematician, e.g. ...
15
votes
2answers
598 views

Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?

A topology on a group is required to be compatible with the group structure (multiplication must be a continuous map $G\times G\to G$ and inversion must be continuous). I've only ever seen the ...
8
votes
3answers
5k views

Difference between “intercept” and “intersect”

What is the difference between intercept and intersect? Can they be used interchangeably? For example, intersecting lines and intercepting lines.
11
votes
3answers
554 views

Has error correction been “solved”?

I recently came across Dan Piponi's blog post An End to Coding Theory and it left me very confused. The relevant portion is: But in the sixties Robert Gallager looked at generating random sparse ...
2
votes
2answers
57 views

Quick criterion to decide whether a limit of functions in $W^{1,p}(\Omega)$ is in that space

Let $\Omega\subset \mathbb R^N$ be an open set. Suppose we are given a sequence $u_n$ in $W^{1,p}(\Omega)$, $1<p\leq+\infty$, such that $u_n\to u$ in $L^p(\Omega)$ and such that $(\nabla u_n)$ is ...
7
votes
1answer
1k views

Why use radical notation instead of rational exponents?

I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, ...
13
votes
2answers
941 views

Is finite group theory still a fruitful area of research?

A while back I was learning about finite groups in an algebra class. I mentioned to a friend that finite group theory might be an interesting area of research to pursue. She asked something along the ...
4
votes
2answers
452 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
1
vote
2answers
325 views

Could you give me some small research questions in the university level?

I want to do some study about math in the university level, but i have no idea about choosing problems. The fact is that i try to take part in a research activity from our school, but when i got the ...
1
vote
1answer
515 views

What is the relationship between variance and energy

I was speaking with someone today who told me that variance, in the sense of probability theory, is equivalent mathematically to energy in physics. Can anyone elaborate on this relationship?
11
votes
1answer
889 views

Which is the newest and most unexplored branch in pure math? [closed]

I recently heard about a branch called tropical geometry developed by Professor Bergman. I was wondering if there´s a newest but yet unexplored math branch, and by newest I mean developed in the last ...
3
votes
3answers
335 views

Why are half-open intervals $(a,b]$ “special” in probability theory?

I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that The Borel $\sigma$-algebra of ${\mathbb R}$ is ...
12
votes
3answers
266 views

definite and indefinite sums and integrals

It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it ...
3
votes
2answers
1k views

Contemporary research in abstract algebra [closed]

I am a Physics undergrad, who is considering the prospects of changing to maths. I was wondering if there is research going on in pure abstract algebra nowadays. I am aware of the fields algebraic ...
3
votes
2answers
716 views

Roadmap for maths self-study for a physics undergrad wanting to pursue research in maths [closed]

I am a second year physics undergrad. After spending about an year exploring mathematical physics (QFT,solitons, etc), I have realized that maybe theoretical physics isnt my passion after all, and I ...
1
vote
2answers
318 views

Linear Algebra presentation

Ok, so let me give you the background story. In my country, unlike the US, there are different high schools for different levels of 'intelligence'. I believe there are 5 or 6 types, the toughest ...
14
votes
5answers
545 views

Inspiring a new generation

Tomorrow is a very special day for both my students and I: we will be starting a calculus course. I'm looking for some nice quotes to read to them to convey just what a complete game changer calculus ...
3
votes
2answers
462 views

Geometric Interpretation of Properties of a Vector Space

I am currently undertaking a self-study on differential geometry and vector spaces, the latter of which is new to me. I was wondering whether the fact that every Vector space has a basis can be ...
2
votes
1answer
885 views

How to work faster on the GRE.

I'm fixin' to go to grad school next year, so lately I've been studying for the mathematics subject GRE. What I have heard, from MSE and other places, is that the most important thing is to learn to ...
4
votes
2answers
984 views

Meaning of the word 'linear' in mathematics

What exactly does 'linear' mean. For example, Hilbert space is a linear space. What is the difference between a linear and a 'non-linear' space. I am starting a self-study of linear algebra so this ...
1
vote
4answers
616 views

Trial and error in maths [closed]

For you, personally, the way you do maths, how much does the way forward seem like trial and error versus absolute? I guess what I mean is that it seems to me that in maths there is kind of a goal ...
2
votes
1answer
762 views

Prerequisite for Differential Topology and/or Geometric Topology

What are the prerequisites to learning both or one of the items? Consider that one will have done some of the "core" classes like Differential Geometry, Real Analysis, Abstract Algebra and POint-Set ...
3
votes
1answer
425 views

Importance of Kripke–Platek set theory

What would be importance of Kripke-Platek set theory? I know that Saul Kripke is one of the most important philosophers in the world, but curious in what place he is in set theory.
8
votes
2answers
320 views

Are competition-type problems useful?

I'm asking from the perspective of an undergraduate. In high school, I found that training with competition problems was immensely helpful. I solve problems on the internet and read solutions. That's ...
5
votes
4answers
278 views

Why is kernel important? [closed]

Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?
1
vote
2answers
378 views

Open courseware for Linear Algebra

I want to self study LA and I have 2 options for open courseware: MIT and khanacademy. Which one do you guys think is more thorough and would you recommend for a beginner? Of course, I also have books ...
5
votes
1answer
730 views

How can I add equations to services such as Keynote or Pages in iPad?

Mathematical formulae are very poor in services such as Windows Word, Keynote, Pages -- or many times totally missing, not to speak about compability -problems. So how can I create and add formulae to ...
12
votes
3answers
1k views

What should a math graduate know? [closed]

There are a lot of undergraduate courses out there and most of them agree on certain things, with regard to the subjects covered. Courses that include mathematics (engineering, physics, etc) are ...
0
votes
3answers
442 views

Project in computer science and mathematics.

My background I'm a third year student. I study mathematics combined with computer science (with focus on modeling, simulations and visualization). In order to get my degree, I have to make a ...
3
votes
1answer
178 views

Order of elements in group

Given $G$ is a group, and $ab=ba$ for $a,b\in G$, we know $o(a)=12,o(b)=18$, in order to calculate $o(ab),o(ab^2),o(a^2b^3)$. Do we need to assume group is a cyclic group in order to use formula ...
4
votes
2answers
401 views

What's the difficulty in finding instantaneous velocity?

I'm reading Stewart's Essential Calculus: EXAMPLE 1 Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ...
11
votes
8answers
447 views

Why is it okay to say two groups are the same, when they're really isomorphic?

This is a pet peeve of mine. Both my textbook and my instructor, often say $G$ is $G'$, when really what they mean is $G$ is isomorphic to $G'$. Why is this an accepted convention?
8
votes
3answers
233 views

value of work in mathematics [closed]

If you're working in a field of mathematics that is not driven by real world problems (or the abstraction of real world problems), then how do you gauge the value of your work? How do you decide what ...
23
votes
2answers
2k views

How do people come up with difficult math Olympiad questions?

The problems that appear in difficult math competitions such as the IMO or the Putnam exam are usually very difficult and require some ingenuity to solve. They also usually don't look like they can be ...
4
votes
2answers
706 views

The usage of ad hoc vs a priori in mathematical papers

I was reading the paper Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, where the author phrased something as this: The analyses of these methods are ad hoc but ...
2
votes
1answer
155 views

Using Sylow theorem without proving:Advisable?

My problem is slightly convoluted: The resource from which I am mainly following, has not introduced Orbit Stabilizer Theorem and have gone for proving it(Sylow) in a highly complex way. In short: I ...
2
votes
1answer
159 views

Why do we care about transposes/duals of linear maps?

I'm just starting to go deeper in my linear algebra education, and this question: Why do we care about dual spaces? helped my intuition and motivation immensely, especially Matt E's answer. So, I'm ...
6
votes
4answers
154 views

Zero polynomial [duplicate]

Possible Duplicate: Polynomial of degree $-\infty$? Today in Abstract Algebra my instructor briefly mentioned that sometimes the zero polynomial is defined to have degree $-\infty$. What ...
37
votes
1answer
1k views

Are difficult exercises good for beginners?

I'm self-studying Rudin's Real and Complex Analysis. I've finished the first two chapters so far, and I don't have any major problems understanding the definitions and theorems. I can prove the ...
2
votes
2answers
139 views

What sums should everyone know?

Well, everyone is an overstatement. Qualifying it: I'm interested in algorithms (complexity) and probabilistic processes/models and I often see sums converge to a certain value in the books I'm ...
2
votes
1answer
205 views

Why don't we have many differential topologist

I am interested in learning differential topology as Milnor, Guillemin, Pollack, Hirsch, Kosinski, etc.. did it. However, I am in University of Toronto as an undergraduate and none of colleges (or ...
10
votes
2answers
1k views

Is time spent going to lectures better spent self studying?

What do you think? I find it more efficient to get a couple of good books, learn by my own, and if i have questions go to class and ask them. Yes, maybe a nice lecture is more illuminating than a ...
3
votes
1answer
354 views

Visualizing Infinity discerning countable and uncountable

This is rather a philosophical question. Although it uses topological notions, it isn't any precise mathematics, so maybe one cannot take it very seriously. Sometimes I try to picture an infinite set ...
4
votes
1answer
396 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
6
votes
3answers
598 views

Mathematics and musical instruments

I want my daughter, who is 6, to be good at maths (of course what she will want will override this). I have heard that people who play musical instruments tend to be good at maths too. I have ...
3
votes
2answers
336 views

Applications of logic in sciences [closed]

i understand math as the study and description of the behavior of mathematical structures, and as you know this math structures could include rings, fields, metric spaces, propositions, categories, ...
3
votes
2answers
579 views

Important implications of Russell paradox?

All of us, maybe in our first incursions on pure mathematics or going further in logic or set theory, need to face the rare and "pretty easy" to understand Russell paradox, but maybe the further ...
4
votes
1answer
139 views

Consequences of Pontryagin Duality?

What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically ...
11
votes
4answers
696 views

Short forms like “haven't”, “don't”, “let's” [closed]

Is it true that short forms like "haven't", "don't", "let's" should not be used in serious mathematical texts?