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3
votes
0answers
129 views

What are the Big Theorems in Information Geometry?

I am working on preparing a talk on information geometry to a young finance/applied math audience. Motivating this area is turning out to be a little difficult. What are some big theorems or results ...
0
votes
0answers
12 views

Language used in projective linear group

In lectures and text on topic of projective linear group, I hear and see the word "factor out" or "quotient out" thrown around a lot. What is the word supposed to mean? If this is vague, I can ...
2
votes
0answers
18 views

Is this an accurate layman's description of the Anti Foundation Axiom

I'm writing an article that covers as one of its topics hypersets/non-well founded sets. In order to do so I have to describe what the anti-foundation axiom (AFA) is my description is currently as ...
345
votes
35answers
41k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
0
votes
1answer
18 views

Multidimensional Cantor diagonal argument for ordering infinite sets [duplicate]

Cantor diagonal argument is a powerful proof technique. It has been used for a lot of proofs. For instance, it has been used to prove that $|\mathbb{N}| < |\mathbb{R}|$. What can we say about the ...
1
vote
2answers
54 views

How to access the world's specialised knowledge? [on hold]

This is a question that relates to almost all domains, I just happen to be passionate about maths. In my early teens, I used to believe that the path through all stages of standard education would ...
66
votes
30answers
31k views

Best Maths Books for Non-Mathematicians [on hold]

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
39
votes
8answers
5k views

How to debug math?

May seem strange as I'm good in programming, but I just started diving into math. ATM I'm learning combinatorics at Khan Academy, and here's an example of a question that I struggled with (that's not ...
2
votes
1answer
134 views

Which numbers are necessary?

The Greeks were initially convinced that all numbers were rational until upon pain of contradiction were forced to accept that $\sqrt{2}$ was irrational and needed to be included in our number system ...
3
votes
1answer
119 views

Soft Question: Why does the Axiom of Choice lead to the weirdest constructions?

I hope this is not too off-topic / soft for math.stackexchange. My basic question is: why does the Axiom of Choice allow for some of the weirdest constructions in math? I'll make a list of the weird ...
39
votes
5answers
4k views

Why do we study real numbers?

I apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\mathbb{R}$ and its subsets (as opposed to any other algebraic ...
2
votes
4answers
86 views

Should we or should we not take $1$ as a prime number? [duplicate]

I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime ...
3
votes
1answer
103 views

Can we characterize the probability generating function as a linear operator?

For a nonnegative integer-valued random variable $X$ with $\mathbb P(X=j)=p_j$, we define the probability generating function of (the distribution of) $X$ by $$P_X(s):=\mathbb E\left[s^X\right] = ...
0
votes
0answers
20 views

What is a probable form of Fermat's triplet?

The mathematical community is now aware beyond reasonable doubt that Fermat's Last Theorem was accurate. However, I cannot help asking, if there were any non trivial Fermat's triplet: -What would ...
74
votes
3answers
6k views

Getting Students to Not Fear Confusion

I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro ...
19
votes
2answers
8k views

How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?

I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school. I'm currently in my ...
1
vote
1answer
15 views

Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
1
vote
0answers
27 views

How zero came to be so useful…(of course it is too elementary !) [on hold]

I have heard on some television documentary(Discovery channel I think) that zero was initially used by an Indian mathematician Brahmagupta(was in fact a priest in Hindu temple) around 650 AD who alone ...
5
votes
1answer
83 views
+50

Any math competitions dedicated to calculations by hand (on a college level)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
33
votes
8answers
7k views

How can I learn to “read maths” at a University level?

When I look at math, it's like my mind goes fuzzy. The only way to describe it is in terms of what I can relate it to. You know how when you read, you see the letters and words, but your brain picks ...
4
votes
1answer
300 views

Is Euler's Introductio in analysin infinitorum suitable for studying analysis today?

I've read the following quote on Wanner's Analysis by Its History: ... our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than ...
1
vote
1answer
22 views

Investigation Problem to challenge mathematical reasoning

I am a sophomore in college and currently enrolled in a Upper Division Problem Solving class. I was assigned a final project in which I need to come up with some sort of mathematical investigation ...
0
votes
1answer
25 views

Computing the “Mean Value” of a Point Sample From an Arbitrary Manifold

A friend of mine noticed that taking the "mean" of two points on the circle isn't as easy as just computing the arithmetic mean of their arguments: If one point has argument $-3.13$ radians and one ...
23
votes
7answers
2k views

Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
2
votes
0answers
24 views

Tricky divergent binomial expansions?

The binomial expansion of $(a+b)^n$, where $n\notin\mathbb{N}$, is given as $$(a+b)^n=a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\cdots$$ In some situations, we can find the result of a ...
1
vote
0answers
46 views

What is the purpose of homomorphisms?

I know that a mapping $\phi:A\to B$ is a homomorphism provided that $$\phi(A*B)=\phi(A)\times\phi(B)$$ where $*$ and $\times$ are two operators on the algebraic structures $A$ and $B$ respectively. In ...
12
votes
7answers
3k views

Should I do all the exercises in a textbook?

The problem sets that you usually get in a university course is a small fraction of the exercises in your textbook. Which raises a question: do you need to solve all the exercises from your textbook? ...
0
votes
0answers
52 views

Am I smart enough for Mathematics? [on hold]

I am 17 and I have been quite interested in Mathematics in the past year. I am thinking about studying Mathematics in college, but I do not know if I am confident enough about my intelligence. I read ...
0
votes
0answers
13 views

Video Lectures on Multi variable Calculus in n dimensions

I am teaching myself multi variable calculus from various resources on the internet. I have completed the famous 18.02 offered at MIT. This is an introductory course on Multi variable Calculus and is ...
2
votes
1answer
46 views

Vladimir Blinovsky's Union-Closed Sets Conjecture Proof

Recently, Vladimir Blinovsky published an article (http://arxiv.org/pdf/1507.01270v6.pdf) claiming that he proved the union-closed sets conjecture ...
5
votes
0answers
62 views
+50

Abstraction and Genaralization

This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me. Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote: from the items $p_{i}$, find a common ...
3
votes
6answers
3k views

Best practice book for calculus [on hold]

I tried with every inch in me to not ask a question such as this but I just couldn't resist asking this. What is the best Calculus practice book? I tried looking around but couldn't find a ...
8
votes
3answers
14k views

Abstract Algebra book with exercise solutions recommendations.

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am ...
2
votes
2answers
82 views

About the Words “recursion” and “recursive”

According to Wikipedia, Recursion is the process of repeating items in a self-similar way. On the other hand, the word "recursive" is an adjective and is often used as a synonym of "computable" when ...
8
votes
1answer
286 views

What is the current state of formalized mathematics?

Russell and Whitehead famously tried to actually create and use a formal system to explicitly develop formal mathematics in their work, "Principia Mathematica." Much more recently, with the aid of ...
2
votes
0answers
60 views

What should I have learnt as an undergraduate? [on hold]

I am getting my bachelor's degree this summer, and I feel like I don't know as much as I should in several fields in math (geometry, functional analysis, measure theory, abstract and linear algebra, ...
16
votes
2answers
904 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
40
votes
11answers
2k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
-2
votes
2answers
118 views

Is there any undergraduate mathematics learning website like edx or coursera?

I want to take courses in mathematics of undergraduate level. But in edx or coursera they have only few undergrad math courses( only calculus or a little bit of algebra). There are no hardcore maths ...
0
votes
1answer
27 views

Simplifying connections between different areas of math [on hold]

Often times I read a brilliant way of making complete sense of a concept in the realm of maths. For example, addition can be built on-top of itself to produce multiplication, and again to produce ...
3
votes
3answers
51 views

What is the motivation for normed division algebras?

The famous Hurwitz theorem states that the only normed division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. What is some good pedagogical motivation why we should think ...
0
votes
0answers
54 views

Example of a collection that is not a set?

Just out of curiosity (the subject is way out of my league at the moment) I have been reading a little about set theory, and I came across Russel's paradox. From what I understood, Russel's paradox ...
0
votes
0answers
4 views

partial order and equivalence relation question [on hold]

Let A = ℤ+ x ℤ+ and R be a relation on A (that is, R ⊆ A xA) defined as follows. (a,b) ~ (x,y) if and only if a + y = b + x. Is R a partial order? Is R an equivalence relation?
3
votes
3answers
44 views

Does the PNT establish a connection between primes and the logarithm?

The prime number theorem states that $$\pi(x) \sim \frac x {\ln(x)}$$ Morally, this seems to suggest that there is a fundamental connection between primes and the natural logarithm. But since we're ...
1
vote
2answers
92 views

Why are all non-polynomial functions are basically exponents?

There's paucity of really "original" functions in Math. Aside from power functions/ polynomials, really the only other function widely used is exponential. For example, $\log$ is simply inverse of ...
8
votes
4answers
1k 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 ...
2
votes
0answers
35 views

Abelianization and analysis of Fundamental Groups

I am working through Hatcher on my own, and currently doing problem $9$ on $p53$. This problem brings up the strategy of abelianization of groups to solve problems of fundamental groups and ...
1
vote
0answers
99 views

My supervisor blocks me from submitting my co-authored paper to a journal [closed]

From the start of my PhD, I had too many disagreements with my supervisor on the research topic (no funding) she gave me. This made our relationship bad and the collaboration unpleasant. In my last ...
0
votes
0answers
35 views

Mathematical conjectures for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...