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0answers
28 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
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0answers
26 views

Integration of a symmetric function

I have a bit of confusion about the following situation. Let's assume that we have a symmetric function $f(x,y)$ where it has the property $f(x,y) = f(y,x)$ for all $x$ and $y$. $x$ and $y$ have the ...
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0answers
17 views

Induction graph theory - dealing with reducing the problem

I have a general question regarding induction in graph theory. Often I am required to use induction in order to prove a theorm. I have seen a lot of cases in which the reduction of the problem was ...
26
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7answers
666 views

Is university math all about proofs?

Do mathematicians do anything else beside writing proofs? It seems like all the "upper-division" math here are about proving something rather than solving for something i.e. instead solving for $x^2 = ...
2
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5answers
223 views

Is there an object in reality that is proven to be uncountable? [on hold]

I've always wanted to come up with a fairly concrete example of an object or realistic set that could be uncountable. Most of the sets I can think about, even the hugest ones, are always countable. ...
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2answers
495 views

How to avoid stupid mistakes in calculus exams without checking the whole process?

Few days ago I failed my Calculus exams. And again it was mostly due to simple mistakes such as forgetting about minus in front of fraction, switching y coordinates of two points etc. The assignments ...
0
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0answers
31 views

Is group theory a generalization of number theory [on hold]

The applications of group theory are abound. Many mathematical objects are examined by associating groups to them and studying the properties of the corresponding groups. But number theory and Graph ...
1
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1answer
31 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?
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1answer
57 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 ...
1
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1answer
51 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 ...
6
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1answer
132 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
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0answers
58 views

Please help? Because I need help? [on hold]

Why does math have numbers????
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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 ...
4
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1answer
72 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 ...
12
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5answers
175 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 ...
0
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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
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4answers
635 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 ...
0
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0answers
27 views

Origin of Matrix A when calculating Eigenvalues and Eigenvectors [closed]

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
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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 ...
1
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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 ...
4
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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 ...
9
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3answers
142 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
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1answer
79 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
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0answers
64 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 ...
2
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1answer
46 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
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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 ...
39
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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 ...
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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 ...
8
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8answers
365 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 ...
27
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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 ...
14
votes
3answers
782 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
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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 ...
13
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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' ...
0
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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 ...
0
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0answers
68 views

Undergraduate 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?
4
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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
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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 $X$, ...
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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 ...
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0answers
44 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
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0answers
47 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
51 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
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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, ...
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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 ...
2
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2answers
49 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 ...
66
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
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0answers
20 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
75 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
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2answers
64 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
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1answer
69 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
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4answers
92 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 ...