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0
votes
1answer
30 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 ...
4
votes
1answer
83 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 ...
14
votes
5answers
209 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 ...
7
votes
3answers
658 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
3answers
46 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 ...
1
vote
1answer
70 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 ...
4
votes
2answers
88 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 ...
9
votes
3answers
167 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 ...
1
vote
1answer
93 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
72 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 ...
6
votes
4answers
213 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 ...
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 ...
40
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 ...
8
votes
8answers
419 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 ...
30
votes
7answers
4k 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 ...
14
votes
3answers
833 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? ...
4
votes
1answer
103 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 ...
12
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' ...
4
votes
2answers
89 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
33 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 $X$, ...
1
vote
2answers
31 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 ...
0
votes
0answers
47 views

How do mathematicians choose which formulas are important?

I'm reading a introductory book on elliptic curves and am having some trouble distinguishing between the important formulas and the insignificant ones. For example, some of the equations introduced ...
4
votes
0answers
52 views

Physical analogies of a math concepts [closed]

In a post Terence Tao explained a very nice way to think about convolution and noted that "one should try to use physical intuition to model mathematical concepts whenever one can". I found this very ...
3
votes
1answer
56 views

What is a functional? And how is it defined for the length?

Im reading about Calculus of varations and there is a lot of references to "the functional" i.e we want to find the minimum of the functional etc. From what i have read, "the functional" is simply the ...
7
votes
4answers
173 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
2answers
96 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 ...
2
votes
2answers
57 views

The difference between the algebraic torus and the geometric torus

I know that the donut-shaped geometric object in $\mathbb{R}^3$ is homeomorphic to a square with identified opposite sites. However, while the latter has a clear symmetry between two dimensions, the ...
68
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
35 views

Linear Algebra and Its Applications Gilbert Strang-Solutions-Unable to find

I am trying to find Linear Algebra and Its Applications Solution Handbook by Gilbert Strang but I am unable to find it any where. I am more focused on this particular book and not anyone else since it ...
6
votes
1answer
89 views

What are the best topics to learn for a first (and second) course in Category Theory?

I am a mathematics student in my last year of undergraduate studies and I'm taking a first Course in Category Theory. The professor that is giving the course is not a category theorist and because of ...
1
vote
2answers
70 views

What I have to do to write and have published an article?

Let's suppose that I have proved a theorem myself and I want to write an article about it, I have a few questions: 1) How do I have to write it ? I mean, what character should I use, what conventions ...
2
votes
1answer
70 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
6
votes
4answers
94 views

$\binom{n}{r}$ versus $^n\mathrm{C}_r$ : which notation is more used?

I know that the notation $\binom{n}{r}$ is more standard to use since we have a $\LaTeX$ command for it while there is no such thing for $^n\mathrm{C}_r$. Now, I'm wondering which notation do people ...
5
votes
1answer
91 views

Suggest a follow up book to Axler's Linear Algebra Done Right?

So I know that a similar question has probably been asked about alternatives or compliments to this book, but I think my situation is different enough to warrant slightly different advice. So I've ...
1
vote
1answer
88 views

Is it ill-advised to read books casually for entertainment? [closed]

I'm a student who has about a year (and a few months) to go before entering a university and I've been reading some math books recently. I'm on Chapter 6 on Rudin's PMA, Chapter 5 in Munkres' Analysis ...
2
votes
1answer
60 views

How to know what kinds of substitution can we do in math?

I have seen in many contexts that somebody out of the blue decides to put $x=y^2$, or $x=t/2$. So how do I know what kind of sustitution I'm allowed to do? Is there any necessary conditions or we ...
2
votes
0answers
76 views

Cognitive processes involved solving IMO level problems [closed]

I am currently 16 years old and, though I'm obviously not as good as most of the people on this site, I have always been considerably better than most of my classmates in mathematics. This, of course, ...
1
vote
1answer
42 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 $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
11
votes
6answers
159 views

Where to find interesting integrals for a Calc III student?

I apologize in advance if this is a very soft question. I won't be surprised or offended if I can't get a good answer. One of my favorite things to do in my spare time, when I'm feeling analytical of ...
3
votes
0answers
40 views

Criterion for Improvement

I was recently asked the question "How do you know when you've become a better mathematician/better at mathematics?," and I realized that at that moment I did not have a valid answer, since I have ...
0
votes
0answers
15 views

Good introductory texts on modular forms/L-functions

I am relatively new to these areas but would like to gain some understanding through an introductory text. I am an undergraduate math major so ideally these books should be accessible to someone with ...
6
votes
0answers
87 views

Is it worth proving every theorem you learn?

Ever since I determined mathematics - mainly set theory and number theory - was my main passion, and I began learning mathematics formally outside of the curriculum posed within secondary schools I ...
6
votes
0answers
94 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
0
votes
1answer
43 views

A problem in understanding principal root in the complex plane.

We know that every complex number has exactly $n$ $n$-th roots in the complex plane, and we usually take (if the context where we are working doesn't tell us more) the one with real and imaginary part ...
2
votes
2answers
76 views

Recommendation of multivariable calculus books

I am looking for some suggestions on a good calculus book I shall keep on hand all the time. I am a graduate student who will be commencing research in the area of theoretical PDE (nonlinear). ...
2
votes
2answers
111 views

Algebra Text Recommendations [closed]

I am looking for any recommendations or suggestions for a good book covering an introduction to the following; Relation , sets and functions, divisibility theory and modular arithmetic , groups, ...
0
votes
1answer
128 views

Latest episode of the big bang theory, vanity card.

I usually don't read these, but this time I did, and this was the card: Does the last mathematical symbols have any meaning? I get that the equal 150.6+V, is there any more meaning behind this?
1
vote
3answers
123 views

Definition of the mathematical proof

How do we define a mathematical proof? Is it a series of arguments? Is it a series of conclusions? Is it manipulation of formulas? Is it a mixture of laws of logic and axioms,theorems or ...
1
vote
0answers
22 views

Prove that $c_1 \phi_1 + c_2 \phi_2$ solves the IVP

Theorem: If $\{ \phi_1, \phi_2\}$ is a fundamental set of solutions of $$x''+p(t)x'+q(t)x=0,$$ then for any initial values $x(t_0)=x_0, x'(t_0)=y_0$, there are constants $c_1$ and $c_2$ so ...
6
votes
1answer
68 views

When vectors act on scalars.

Background. I've been struggling through an introduction to differential geometry this semester. Recently, a tiny part of what we've been learning "clicked" for me, and to solidify this, I'd like ...