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1answer
54 views

Inverse functions multivalued or not?

The square root of $y$ is usually defined as the positive solution $x$ to $y=x^2$, so the negative variant is not considered. In the same way, the inverse cosinus and sinus give the solution on ...
2
votes
2answers
62 views

Why does unary minus operator sometimes take precedence over exponentiation, and sometimes it doesn't?

How should I evaluate 2*-2^3? Which one of these two is the correct one? 2*((-2)^3) 2*(-(2^3)) I was wondering what was ...
1
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2answers
45 views

How to improve visualization skills (Graphing)

Okay, so my problem is, that I have difficulty visualizing graphs of functions. For example, if we have to calculate the area bounded by multiple curves, I face difficulty in visualizing that how the ...
18
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1answer
285 views

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
2
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1answer
34 views

What are the most important corollaries/consequences and applications of certain algorithms in elementary number theory? [closed]

What are the most important corollaries/consequences and applications of Division Algorithm, Euclidean Algorithm and Fundamental Theorem of Arithmetic? I've been studying Elementary Number Theory for ...
2
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0answers
19 views

Applications of module's length

I'm studying some theory about module's length and want to know motivation for this definition. I know that it's useful for intersection theory, but i know only one example from intersection theory: ...
9
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2answers
236 views

Why is it so difficult to find beginner books in Algebraic Geometry?

I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...
1
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0answers
40 views

Algebraic surfaces in the language of scheme

Are there materials(lecture notes, books...) that deal with algebraic surfaces in the language of schemes? I am not good at/familiar with the analytic way, and also prefer the scheme-theoretic ...
4
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1answer
145 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
44 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
31 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 ...
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8answers
1k views

Is university math all about proofs? [closed]

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
votes
5answers
259 views

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

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
601 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 ...
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1answer
34 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?
0
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1answer
73 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
62 views

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

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
votes
1answer
150 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
0
votes
1answer
35 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
89 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
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5answers
227 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
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3answers
664 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 ...
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3answers
54 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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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 ...
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2answers
95 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
183 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
101 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
75 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 ...
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4answers
221 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 ...
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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 ...
8
votes
8answers
487 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
884 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
115 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 ...
4
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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
36 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
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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 ...
4
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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
58 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
187 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
99 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
63 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 ...
69
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 ...
6
votes
1answer
90 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 ...
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2answers
72 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
71 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
97 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
109 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 ...
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1answer
97 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 ...