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7
votes
2answers
1k views

where did determinant come from? [duplicate]

Possible Duplicate: What's an intuitive way to think about the determinant? I just learned the basics of matrices. Then I came across the magical formula $$\det(AB)=\det(A)\det(B)$$ I ...
4
votes
1answer
7k views

What is the prerequisite knowledge for learning discrete math?

To become a better computer programmer I would like to take the time to learn discrete mathematics, but I am positive that I do not have the required existing knowledge to do so. So I would like to ...
24
votes
4answers
987 views

Are there any common practices in mathematics to guard against mistakes?

It occurred to me that math is somewhat like programming (or vice-versa, if you prefer) because, in both, it is easy to make mistakes or overlook them, and the smallest error or misguided assumption ...
8
votes
2answers
418 views

Mental processes while doing math

This is a soft question concerning the mental processes at work when doing maths. I hope this question is not too vague, and I believe I haven't found a similar one previously posted on MathSE. In ...
23
votes
2answers
1k views

Efficient ways to read and learn a new topic [closed]

I started reading the book "Topology without tears" by Sidney A Morris and lecture notes on "Elementary Number Theory" by WWL.Chen. To get the maximum out of the book and understand the material ...
49
votes
17answers
26k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
5
votes
2answers
334 views

Is Complex plane more than just a topological space with $\mathbb{R}^2$ topology?

The Set of Complex numbers is a field as well as a nice topological space homeomorphic to $\mathbb{R}^2$. But why such a particular interest for this space? For instance what is more special about it ...
5
votes
2answers
686 views

Can one write a PhD thesis before going into a PhD progam? [closed]

Let's say I write a PhD thesis by looking at papers on arxiv. Can I use this as my PhD thesis one I enter into the program? In other words, can you write a PhD thesis before going to graduate school? ...
6
votes
6answers
2k views

Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real ...
5
votes
1answer
267 views

What's the difference between Ramsey theory and Extremal graph theory?

Wikipedia teaches us that problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?" It ...
0
votes
3answers
523 views

“Mathematical Induction”

I realize this question borders on not qualifying as answerable or mathematical enough, but I would suspect it relevant somehow. I'll remove it if it's not. If you look at some explanations of ...
1
vote
1answer
531 views

Lucas' Theorem and Pascal's Triangle

I have a general question about Lucas' Theorem. Lucas' Theorem says the following: Theorem (Lucas' Theorem) Let $p$ be a prime number. Write $n$ and $k$ in base $p$: $n = a_0 + a_{1}+a_{2}p^{2} + ...
1
vote
1answer
218 views

Social network representation: graphs or sets?

for social network representation, what is better, sets or graphs ? What kind of feature the first gives that the second doesn't and viceversa?
4
votes
1answer
371 views

How to understand currents in geometric measure theory?

I find it is hard to catch the current, sometimes it is just the picture as its support set (if I do not miss it). What is the heart idea of the current? What are the benefits to introduce such an odd ...
6
votes
3answers
902 views

Rigor in Mathematics

Since starting my undergrad studies (in Maths -- !) I have struggled to grasp the concept of "rigor". My "proofs" tend to have the right framework but somehow they are said to "lack rigor", this has ...
3
votes
0answers
284 views

Differential Forms

I just had a general question about differential forms. Background. Let $f = f_k$ and define $f^{k-1}$ on $I^{k-1}$ by $$f_{k-1}(x_1, \dots, x_{k-1})= \int_{a_k}^{b_k} f_{k}(x_1, \dots, x_{k-1}, x_k) ...
4
votes
0answers
127 views

Status of videos from the Arf-Kervaire invariant problem conference at MSRI

This is not a real maths question (math-related though). A couple of years ago Mike Hill, Mike Hopkins and Doug Ravenel solved the Arf-Kervaire invariant problem. There was a conference at the MSRI ...
1
vote
1answer
404 views

What does “a set of things” mean?

Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean: Set inclusion) ...
5
votes
1answer
1k views

Can I go through Hartshorne without knowing much analysis?

I know intro abstract algebra and some real analysis. Is this enough to study algebraic geometry from the book of Hartshorne?
7
votes
3answers
757 views

Where can one find (freely, online) mathematical articles with a fighting chance to be understood by high school students and undergraduates?

I am an undergraduate non-math major. I just finished my university's engineering calculus series, looking forward to linear algebra in the coming semester. To be frank, I always despised math because ...
12
votes
6answers
3k views

Advice for Self-Study

I am a senior in high school who has taught myself through Calculus BC and I got a 5 on the exam. However, I have taken all the math I can at my school. I have also taught myself multi-variable ...
2
votes
4answers
9k views

How to calculate Maximum or Minimum of two numbers without using if?

How to to calculate the maximim or minimum of two numbers without using "if" ( or something equivalant to that manner)? The above question is often asked in introductory computer science courses and ...
14
votes
2answers
567 views

open conjectures in real analysis targeting real valued functions of a single real variable

I am hoping that this question (if in acceptable form) be community wiki. Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may ...
6
votes
0answers
301 views

homology groups, physical interpretation

One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are ...
7
votes
5answers
2k views

What is Fourier Analysis on Groups and does it have “applications” to physics?

I am trying to be as specific as possible, but I am extremely unclear about this topic (Fourier Analysis on Groups). In Reed-Simon Vol II (Fourier Analysis, Self-Adjointness) there is some ...
11
votes
5answers
1k views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
2
votes
1answer
197 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
697 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?
44
votes
3answers
9k 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
215 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
810 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
173 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
327 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
224 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
495 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
533 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
832 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 ...
5
votes
3answers
253 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
328 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
506 views

Advice about a career interest in Mathematics [closed]

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
882 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 ...
16
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 ...
16
votes
0answers
497 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
446 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 ...
50
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
360 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 ...
14
votes
3answers
885 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
410 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, $\aleph_0$ ...
107
votes
10answers
6k 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 ...