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2
votes
1answer
192 views

Embrace applied mathematics [closed]

Does anyone have any suggestions as to what is a good topic for a short talk on theoretical physics to a bunch of Math and Physics undergrads that might make them "embrace" theoretical physics? ...
1
vote
1answer
650 views

Whats the difference between arithmetic geometry and algebraic geometry?

both seem to be about geometry. why the distinction? I mean which preceded the other? Why is algebraic geometry more popular?
43
votes
3answers
8k views

What is the importance of Calculus in today's Mathematics?

For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
1
vote
2answers
209 views

Are there any applications of Fourier series/analysis in General relativity

I'd like to know if there are any applications of Fourier analysis / Fourier series expansion in General relativity ? I mean how Fourier transform has applications in Quantum mechanics.
20
votes
2answers
723 views

Publication Quality Mathematics Diagrams

I am curious what sort of applications people use to make very nice diagrams that often appear in papers and books. I attached an example of the sort of diagram that I am interested in making, ...
3
votes
2answers
168 views

Markov Chains and Linear Transformations

I just have a quick question about Markov Chain and linear algebra. Background. Let $\{M_n: n= 0, 1, 2, \dots \}$ be a Markov Chain. We can represent the transition probabilities $_{n}Q^{(i,j)}$ in a ...
8
votes
1answer
312 views

Categorification of characteristic polynomial

It's probably just my applied background talking, but I'm puzzled by characteristic polynomials of matrices. Useful things like that are usually closely connected to some nice functor or homomorphism, ...
2
votes
2answers
212 views

Survival Functions and PDFs

I have a general question about survival functions and their associated PDFs (probability density functions). Background. A survival function $s(x)$ is the probability that an individual will survive ...
3
votes
1answer
439 views

Making theorems into flashcards?

I am studying real analysis and trying to convert some of the rather wordy theorems into flashcards. Some of the theorems have names and so it's easy to make a flashcard that just asks that I state ...
6
votes
2answers
487 views

on the generic points of a scheme

This question may be a little bit metaphysical:are there any important properties about the generic points on a scheme?Or rather,why do we introduce the concept of generic point?I am not very clear ...
6
votes
5answers
788 views

Materials for self-study (problems and answers)

I'm hoping to self-study Geometry, Algebra, Calculus, Vector Calculus, Linear Algebra, Probability and Statistics, and other intermediate maths. I've found the best way for me to learn is to work on ...
4
votes
3answers
243 views

What is the motivation for the definition of the expected value?

I have a general question about expected values: For a discrete random variable, $$E[X] = \sum_{i=1}^{\infty} x_{i}p_{i}$$ and $$E[X] = \int_{-\infty}^{\infty} xp(x) \ dx$$ for a continuous ...
2
votes
1answer
310 views

Geometric Interpretation of Complexified Tangent Vectors on a Real Manifold

What is a good geometric way of thinking of complex tangent vectors on a manifold? I can convince myself that I understand tangent vectors by thinking of them as paths on the manifold. Is there a nice ...
1
vote
1answer
482 views

Advice about a career interest in Mathematics

I am a graduate student and am into 2 years of PhD. My current specialization is in signal processing. During this period, in my spare time, i came up with an idea which seems to be a research ...
12
votes
2answers
826 views

How does one know that a theorem is strong enough to publish?

Question. How does one know that a theorem is strong enough to publish? Basically, I have laid out a framework in which many theorems may be proven. I'm only 18 and therefore lack knowledge of ...
1
vote
0answers
117 views

Does number theory have any role in the proof of convergence of Fourier series for certain functions?

Does number theory have any role in the proof of convergence of Fourier series for certain functions? I vaguely remember reading in a book on signal processing, way back, that the proof (original ...
13
votes
4answers
2k views

How to propose a conjecture

What are the basic things (about when and how) to be kept in mind while proposing a conjecture in Mathematics. Should it accompany solid efforts at proof. When should any one think of proposing a ...
13
votes
0answers
440 views

Irrationality of e

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
6
votes
1answer
386 views

Why study only rook polynomials?

In introductory combinatorics, there is an emphasis on rook polynomials. But what is the significance of only considering rook polynomials? Why not consider "knight polynomials" or "bishop ...
47
votes
2answers
3k views

Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
4
votes
1answer
327 views

Looking for an “arrows-only” intro to category theory

I have often seen it remarked in passing that the "collection of objects" that appears in the standard definition of a category is, strictly speaking, superfluous, and that it is possible to give an ...
12
votes
3answers
784 views

Looking for student's guide to diagram chasing

I'm teaching myself some category theory, and I find that I'm very slow with diagram chasing. It takes me some times a very long time to decide whether adding an arrow to a diagram preserves the ...
3
votes
2answers
395 views

Comparing infinite numbers

Suppose you have 2 infinite numbers, say $A$ and $B$. $A$ is an element of the hyperreals, so that $A$ is greater than every real number. $B$ is the size of the set of natural numbers, $\aleph_0$ ...
98
votes
9answers
5k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
2
votes
2answers
475 views

What is the connection between Chinese remainder theorem and Lagrange interpolation?

In Surprising Generalizations, it is mentioned that Chinese remainder theorem and Lagrange interpolation are specific instances of the same thing, my question is what is their common ...
14
votes
5answers
1k views

What does “formal” mean?

I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...
10
votes
2answers
1k views

Ideas of finding counterexamples?

The questions are from an exercise in Gibert Strang's Linear Algebra. Construct $2$ by $2$ matrices such that the eigenvalues of $AB$ are not the products of the eigenvalues of $A$ and $B$, and ...
1
vote
1answer
236 views

What are some of the factors that go into ranking a math department? [closed]

In terms of research, how do organizations like the US News World report compare math departments from US universities? Do they look at the range of mathematical research and the depth? Do they look ...
2
votes
3answers
347 views

Should I write out stuff? [closed]

When I go through textbooks should I write out solutions to the exercises? Or is it fine if I just do it in my head? I mean either way you are still doing the problems right?
31
votes
3answers
3k views

How do people apply the Lebesgue integration theory?

This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas. In a lot of calculus textbooks, there is ...
7
votes
5answers
281 views

Beside transcendental or uncomputable numbers what other types of numbers are there?

What other types/categories of numbers are there that we know of today (i.e. some one has done some work on them, like Chaitin's uncomputable $\Omega$ number)? Of course there are uncountably many ...
9
votes
4answers
1k views

What is the name of the vertical bar?

I've always wanted to know what the name of the vertical bar in these examples was: $f(x)=x^2+1\mid_{x = 4}$ (I know this means evaluate x at 4) $\int_0^4 (x^2+1) dx = (\frac{x^3}{3}+x+c) ...
7
votes
2answers
130 views

Why is $G(k)$ “more fundamental” than the Hilbert-Waring function $g(k)$?

In the Wikipedia entry for Waring's problem, the section on $G(k)$ starts as: “From the work of Hardy and Littlewood, more fundamental than $g(k)$ turned out to be $G(k)$, which is defined...” There ...
4
votes
1answer
236 views

Which Fourier transform should I use in PDE?

I learned two "different" Fourier transforms in the real analysis course. For any function $f\in C({\bf R/Z;C})$, and any integer $n\in{\bf Z}$, we define the $n$th Fourier coefficient of $f$, ...
29
votes
8answers
2k views

Partial differential equations in “pure mathematics”

One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in Wikipedia follow this pattern. I can't help wondering whether there are ...
1
vote
2answers
300 views

How to pronounce ℓ? Is there any other way than 'ell'?

Reading aloud from a graph theory textbook, I encountered the letter ℓ. I assume that most people simply pronounce it 'ell'. However, is there any other way to pronounce it?
6
votes
2answers
679 views

Greek Alphabet - Pronunciation [closed]

I'm studying computer science and I'm starting to come across a lot of maths! I thought it would be fun (yes I really am a geek!) to learn the greek alphabet as it used in mathematics. I was wondering ...
5
votes
5answers
360 views

Do addition and multiplication have arity?

Many books classify the standard four arithmetical functions of addition, subtraction, multiplication, and division as binary (in terms of arity). But, "sigma" and "product" notation often writes ...
3
votes
0answers
210 views

Designing a mathematical physics class

Surprisingly, the university (a major tech school) I attended does not offer a mathematical physics class. Consequently, I often get asked by my physics friends what are some good math classes to take ...
8
votes
3answers
716 views

Best way to set up a wiki for maintaining a structured math journal

Does anyone know of a tool which Can display formulas neatly, preferably like this website without hassle. (Unlike wikipedia with :<math>) Has a wiki like ...
5
votes
3answers
760 views

Understanding of the Mean Value Theorem in PDE

I learned the following theorem in Folland's Introduction to Partial Differential Equations(p.69 Chapter 2): Suppose $u$ is harmonic on an open set $\Omega\subset{\mathbb R}^n$. If $x\in\Omega$ ...
6
votes
6answers
965 views

Why some people don't like proofs by contradiction [duplicate]

Possible Duplicate: Are the “proofs by contradiction” weaker than other proofs? I have been active on this site for two months and on a few occasions I noticed that some people ...
26
votes
4answers
1k views

Geometry or topology behind the “impossible staircase”

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the ...
12
votes
2answers
894 views

Math Notes and Knowledge Organization Methodology

I, like many of you I suspect, take copious notes when reading and working through math exercises/theorems/constructions. I have stacks and stacks of notes ranging from one day old to several years. ...
73
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 ...
7
votes
5answers
2k views

Experiences with Rudin?

So I am trying to tutor a friend in analysis. This is her first time with proofs. We are on chapter 2 – the topology chapter – of Rudin's Principles of Mathematical Analysis and ...
7
votes
3answers
722 views

Are all numbers real numbers?

If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego ...
1
vote
1answer
143 views

Appropriate book for propositional logic

I am not looking for a good book but an appropriate book that is suitable for my logic course. Currently the professor only offers lectures. (Not sure why, perhaps there is no universal approach to ...
1
vote
1answer
174 views

online classes for math masters pre reqs

I'm currently getting my masters in video game development and next year want to start on my math masters. Some of the pre-reqs I need to still take are calc 3 and linear algebra, does anyone know of ...
28
votes
4answers
783 views

How should I approach taking math tests?

I always do bad on all my math tests yet I do great on projects and homework. Also I like doing research. For exams that have proof based questions I just freeze up under pressure. I just can't do ...