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6
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7answers
1k views

A book for abstract algebra with high school level

Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on ...
6
votes
4answers
5k views

What is more elementary than: Introduction to Stochastic Processes by Lawler

I have trouble to reading this book! What book is more elementary/preliminary than this book: Introduction to Stochastic Processes by Lawler
1
vote
5answers
17k views

Rules for Product and Summation Notation

When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} (2 + 3i) = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + ...
28
votes
5answers
1k views

What does it mean when two Groups are isomorphic?

I'm not asking for the formal definition I know it. An isomorphism is a bijective homomorphism. In my book it's indicated many times when two groups are isomorphic, and I don't understand what's the ...
27
votes
4answers
7k views

What does it really mean for something to be “trivial”?

I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise ...
25
votes
3answers
2k views

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$(\forall a,b\in L) x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet ...
23
votes
7answers
12k views

Teaching a 4 year old maths

Im 18 years old and getting to grips with advanced mathematics (pre-university) and I have a younger brother of 4 years old (quite an age gap). I want to get him interested in learning (and away from ...
22
votes
1answer
2k views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
19
votes
2answers
499 views

A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...
18
votes
1answer
958 views

Why should a combinatorialist know category theory?

I know almost nothing about category theory (I have just skimmed the first chapters of Aluffi's algebra book), reading this question got me thinking... why should someone mostly interested in ...
15
votes
5answers
977 views

In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
15
votes
2answers
2k views

Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
14
votes
5answers
1k views

I need help finding a rigorous precalculus textbook

I dislike modern textbooks; their cookie-cutter approach and appearance, over reliance on breaking things down into little boxes, the general spoon-feeding they engender and most of all the poor ...
14
votes
4answers
6k views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
14
votes
7answers
2k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their ...
14
votes
8answers
6k views

How would you describe calculus in simple terms?

I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?
13
votes
6answers
541 views

Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like ...
13
votes
6answers
2k views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
13
votes
5answers
221 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
13
votes
3answers
1k views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
13
votes
1answer
765 views

Compact = Closed + Bounded + (?)

In $\mathbb{R}^n$ we know (Heine-Borel Theorem) that a set is compact if and only if it is closed and bounded. In $C(X)$ for a compact metric space $X$, we know (corollary of Ascoli-Arzela Theorem) ...
13
votes
5answers
2k views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
11
votes
2answers
708 views

How can I pick up analysis quickly?

I have a 2-3 week recess from university for winter break. In this time, I would like to learn analysis, starting with Walter Rudin's Principles of Mathematical Analysis, and then, if at all possible, ...
9
votes
5answers
3k views

How do you rebuild your Math skills after college? [closed]

I minored in Math in college (B.S. Software Engineering), and went through advanced Calc, Differential Equations, etc. But about 6 months after I graduated I had lost all my formulas. Now, about 5 ...
9
votes
1answer
462 views

Good references for stacks

I have seen stacks come up in various settings recently. I understand that, at least heuristically, they are some sort of generalization of a scheme, but I don't actually know anything about them. ...
7
votes
3answers
970 views

Exercise books in analysis

I'm studying Rudin's Principles of mathematical analysis and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as a companion to ...
41
votes
2answers
5k views

What does “communicated by” mean in math papers?

[This question involves mostly math papers, and may be relevant to graduate students learning to write and cite papers, although this is my only justification for this being a math question.] Usually ...
37
votes
6answers
1k views

Need a result of Euler that is simple enough for a child to understand

Talking to my 8 yr old about "the greatest mathematician of all time", I said it was probably Gauss in my opinion, but that Gauss was not very kind to his kids (for example, forbidding them to go into ...
19
votes
5answers
1k views

Results that were widely believed to be false but were later shown to be true

What are some results which were widely believed to be false, but were later to be shown to be true, or vice-versa?
18
votes
11answers
1k views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
18
votes
1answer
2k views

Strategies and Tips: What to do when stuck on math?

Since math students will be stuck on some math at some point, what strategies or tips can help (to assuage this recurrent reality of maths)? Certainly, this wonderful website helps; here: one can ...
17
votes
1answer
1k views

Arxiv - what should I expect

I went through arxiv.org primer and some questions at MO and here about arxiv. There are many of them so I could have missed answers to some of my questions. My apologies in that case. I have never ...
16
votes
1answer
243 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
15
votes
9answers
644 views

Definite integrals with interesting results [closed]

I just stumbled across the fact that $\int_{-\infty}^{+\infty}{e^{-x^2}dx}=\sqrt{\pi}$. This intrigued my already-existing interest in integrals. It made me wonder, are there other integrals with ...
15
votes
3answers
2k views

History of elliptic curves

In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
14
votes
4answers
3k views

mathematical maturity

So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel ...
13
votes
3answers
276 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
13
votes
3answers
656 views

Where are the geometric figures in those “advanced geometry” textbooks?

It is me again to bother you. Since my last post I started to look at some seemingly "serious" mathematics for background study. Accidentally I went into a university bookshop and came across some ...
11
votes
3answers
6k views

Baby Rudin vs. Abbott

I am considering Stephen Abbott's Understanding Analysis and Walter Rudin's Principles of Mathematical Analysis. I am looking for a comparison between the two that addresses both of the following ...
11
votes
2answers
1k 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 ...
11
votes
3answers
605 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
11
votes
2answers
641 views

Category Theory with and without Objects

Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an ...
10
votes
0answers
194 views

Is Category Theory geometric?

In "From a Geometrical Point of View" (http://www.amazon.com/gp/aw/d/1402093837?pc_redir=1407132421&robot_redir=1) Marquis states that category theory is thoroughly geometric. Could someone ...
10
votes
5answers
2k views

Advice for benefits to directly use analysis textbook to replace calculus

Main purpose: For self-learning performance, neither for exam nor degree courses. Calculus textbook using now[1]: Calculus I, Weinstein&Marsden, UTM, Springer Question Description: I've been ...
10
votes
1answer
1k views

What distinguishes the Measure Theory and Probability Theory?

It is clear that the Theory of Probability works primarily with limited measures on measurable spaces. On the other hand there is a folklore that says that what distinguishes Measure Theory and ...
9
votes
5answers
629 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
9
votes
1answer
353 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
9
votes
2answers
896 views

Local vs. global in the definition of a sheaf

Apologies in advance that this question is inescapably soft. What I am stuck on is squishy; I have the feeling that if I could even make it precise, I'd already be satisfied. To what extent is a ...
9
votes
4answers
455 views

Beside transcendental or uncomputable numbers what other types of numbers are there?

What other types/categories of numbers are there that we know of today (i.e. some one has done some work on them, like Chaitin's uncomputable $\Omega$ number)? Of course there are uncountably many ...
8
votes
1answer
2k views

Very interesting graph!

I found a VERY interesting graph on http://www.xamuel.com/graphs-of-implicit-equations/. It looked very, very cool, but the equation of the graph is so simple! Here is the image of the graph: My ...