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0
votes
0answers
8 views

Exceptions in functions

I have recently started studying functions(topics such as periodicity, odd/even, into/onto, etc.). I was wondering if there are any strange exceptions to the general rule that is taught?
8
votes
2answers
152 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
74
votes
22answers
2k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
12
votes
5answers
172 views

Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?

Is Algebraic Number Theory the study of the theory of algebraic numbers? Or is it the study of the theory of numbers from an algebraic viewpoint? Or is it both? I know I can just find a wiki article ...
2
votes
3answers
545 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
0
votes
1answer
51 views

Severe problems with math undestanding

Recently (although still in high school) I've been at university, more precisely at information science engineering as apprenticeship. I want to become an operating system programmer but I severely ...
20
votes
10answers
416 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. For example: When I read from the book Proof from the Book, I saw there were ...
6
votes
1answer
129 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
1
vote
1answer
50 views

Is Mathematics a branch of “Natural Science”? [on hold]

Actually, I was seeking for top universities, which has mathematics depart, in Pakistan and I found one, namely Quaid-i-Azam University. Which is known for its Education in "Natural Science". Then I ...
9
votes
2answers
3k views

Geometric intuition behind gradient, divergence and curl

I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
16
votes
2answers
177 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
9
votes
3answers
140 views

What do groups and rings “look like”?

Taking undergraduate physics courses, I had to deal with Euclidean vectors often. In classes like Calc III, the concept was also there. I'm not sure if this is why, but I've always had a more ...
4
votes
1answer
52 views

Derived functors - homotopical vs homological approach

In a first course in homological algebra, the lecturer introduced derived functors as universal $\delta$-functors, whose universal property is splicing short exact sequences into long ones. It so ...
0
votes
1answer
25 views

The difference between a matrix valued random variable and an $n \times p$ matrix of data

So I am totally new to the field of random matrices, but I was not sure about how they are applied. According to Wikipedia, a random matrix is "a matrix-valued random variable—that is, a matrix some ...
-14
votes
0answers
58 views
9
votes
3answers
204 views

Is Adobe Acrobat's icon a special function?

It looks like a function in polar coordinates. Is it a special function ?
1
vote
1answer
36 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
4
votes
1answer
69 views

Is general topology essential for applied mathematicians?

I am a second year undergraduate college student interested in applied math program. I hear a lot that general topology(e.g. the first half of Munkres' book Topology) is very useful, but is it really ...
7
votes
0answers
150 views
+100

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a sense, unorganized (see Open problems in ...
0
votes
2answers
56 views

Seeking advice from all [on hold]

I've come back to education after 4 years and I feel very out of practice, currently I am studying a-levels and need to pass with excellent grades for my ill fathers sake as it is his last wish. I am ...
7
votes
4answers
633 views

What is the motivation for complex conjugation?

I have been dealing with complex numbers for few years now. But when I've tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with. For a ...
1
vote
0answers
60 views

Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...
65
votes
10answers
3k views

Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting ...
0
votes
0answers
27 views

Origin of Matrix A when calculating Eigenvalues and Eigenvectors [on hold]

I understand how to calculate Eigenvalues and Eigenvectors ($Ax = \lambda x$), but what I don't understand is how the matrix $A$ originates. Is its origin from measurements or the like? Thanks in ...
1
vote
3answers
36 views

What book is good in studying beginning optimization?

Recently, I heard some talks about Optimization. And I am beginning to love that field. I want to study beginning optimization, what book can you recommend for me? Also what tips can you give to a ...
14
votes
7answers
4k views

Choosing a text for a First Course in Topology

Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
4
votes
2answers
63 views

Modeling curves in nature?

On my windowpane, I've traced the contour of a distant line of hills as they appear to an observer sitting in the sill. This short curve can of course be viewed as a continuous and single-valued ...
7
votes
4answers
148 views

Can we still learn from the old masters?

So, let me first describe how my doubt originated: out of curiosity I started to study Newton's Opticks, a book written more than 300 years ago. I was doing some of the experiments described on it, ...
1
vote
1answer
78 views

Difference between maths in physics degree and maths in a maths degree

I asked this question on the Physics site, but it got closed, so I'll try here. Basically, I was wondering what are the main differences between the maths you learn in a mathematics degree and the ...
8
votes
0answers
61 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
38
votes
4answers
4k views

What's wrong with l'Hopital's rule?

Upon looking at yet another question on this site on evaluating a limit explicitly without l'Hopital's rule, I remembered that one of my professors once said something to the effect that in Europe ...
3
votes
4answers
135 views

What is the smallest positive integer that has never been mentioned?

The set of positive integers is infinite. The set of explicitly mentioned positive integers is finite. Therefore, there is a non-empty set of positive integers that have never been mentioned, and ...
4
votes
6answers
1k views

“Vectors aren't really numbers” - how sound is that statement?

Since I first learned about vectors, I noticed something interesting: almost any numeric formula can be replaced by a vectorial formula by just replacing addition, multiplication, etc., with their ...
2
votes
1answer
42 views

Examples of arguments from connectedness

Suppose $X$ is a connected topological space. A typical way that we prove a property $P(x)$ holds for all $x \in X$ is to show that $P$ is an open and a closed condition, and that $P(x)$ for some $x ...
-2
votes
2answers
50 views

Obvious or not? [closed]

Motivated from this question. I'm looking now for examples of theorems/propositions/Lemmas etc. in different ares of mathematics that are obvious at the first sight, but then turned out to be hard to ...
14
votes
3answers
780 views

Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...
11
votes
2answers
975 views

What would be a good way to memorize theorems about algebra?

This post is not constructive, so maybe this rather should be posted on CW, but since there is a 'soft-question' tag, i'm posting it here. ===================== I believe the best way to memorize ...
26
votes
7answers
3k views

Are there mathematical objects that have been proved to exist but cannot be described in words?

This might be a very stupid, and possibly philosophical question, but attempt to apply mathematics to everything plus inspired by this question caused me to ask this question Is there any ...
8
votes
8answers
358 views

(Soft) What maths should I concentrate on at 16-18 years old? [closed]

Some background information first of all: I'm 16 now and just started studying mathematics intensely. I live in the UK and my goal is to eventually become very good at advanced mathematics (graduate ...
13
votes
6answers
2k views

Can a 18 year old high school student publish a paper?

Is it possible for an 18 year old high school student to publish a maths paper in a journal? The title of my paper is 'Complex structure of the sixth dimensional sphere from a symmetrical fracturing' ...
1
vote
2answers
93 views

A report about complex numbers

I was told to make a report for mathematics, and I could choose my own subject. I chose complex numbers, because I really think they are interesting. However, my teacher says that there isn't a lot of ...
4
votes
3answers
3k views

What are the real-world applications of real analysis?

I've read the wikipedia article on mathematical analysis and this, but I can't exactly find an answer. Is real analysis just some pure math, or does it really have something to with physical ...
0
votes
1answer
46 views

Gap in Math Education [closed]

My graduate school applications fell through and I was wondering if there was a job I could do that would keep me improving as a mathematician, or if the best I could hope for is the least invasive ...
6
votes
2answers
226 views

State of the progess of the automated proof checking

I recently came across a concept of automated proof checking. I am very intrigued by the idea, that in the future all the proofs could be verified by a computer. Moreover, some proofs were already ...
0
votes
0answers
61 views

Unergraduate Research review. [closed]

I am(an undergrad student) looking for online forum, group (or something like that) to discuss/review my ideas.Any suggestion ?? where I can discuss my ideas?
0
votes
2answers
119 views

Choosing a topic for a short talk in algebraic topology [closed]

I'm attending my second course in Algebraic Topology. Exam consists in preparing (and present) a talk. Course's arguments are classical results of Homotopy Theory. I had selected "Homotopy is not ...
4
votes
2answers
81 views

Is there a name for the $n$ in $\mathbb{R}^{n}$ in general?

How to call the $n$ in $\mathbb{R}^{n}$ in general? It is cumbersome to say something like $n$ is the number of the folds of $\mathbb{R}$ in the Cartesian product ... If $\mathbb{R}^{n}$ is regarded ...
0
votes
0answers
24 views

Random Variables: Expected Values and Values computed from a data

Given a random variable $X : \Omega \rightarrow \mathbb{R}$, we have $E[X]$, which is called expected value of the random variable. I have one random variable $X$, ...
14
votes
10answers
2k views

Math problems that are impossible to solve [closed]

I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering ...
1
vote
2answers
29 views

Dividing finite numbers by infinite numbers

I am no great mathematician but I have a question which I can't seem to find a answer for. How can one divide a finite by a infinite number? For example if you have a circle with a circumference of ...