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2
votes
1answer
37 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
3
votes
2answers
94 views

What's the significance of defining group as a group object in category $\mathcal{Set}$?

At first sight, redefining group as a group object in the category of sets $\mathcal{Set}$ seems just like a meaningless restatement, but when we apply this definition to other categories, ...
1
vote
2answers
66 views

Should the theory be studied thoroughly before solving exercises?

Most of the books claim in the preface that the important part of the book is in the exercises, which makes sense considering that solving problems improves in great depth the understanding of the ...
0
votes
0answers
72 views

math student looking to do better in math competitions.

I am currently in my summer vacations. Next year I will star my undergraduate studies in mathematics. I used to be in mathematics competitions. Last year I got a silver medal in my countries national ...
1
vote
2answers
53 views

Formalizing the Fallacy of Composition

Is there a well-known formalization of the fallacy of composition? More generally, where in mathematics is it true that if a property holds for all of some elements of a set it holds for the whole ...
2
votes
0answers
62 views

“Teach yourself” guides [closed]

I really liked Teach Yourself Logic: A Study Guide by the user Peter Smith. It is a thorough guide how to teach yourself logic and set-theory from scratch up to any level with book recommendations for ...
3
votes
1answer
111 views

How should one go about obtaining “mathematical maturity”?

tl;dr: Is mathematical maturity better obtained by doing hard subjects slightly out of your reach, or by doing more simple subjects to gain experience? The end of the semester is close, and I ...
2
votes
2answers
59 views

Deciding which questions to do in a maths exam

I wasn't quite sure if this was the best place for this question. If faced with a maths exam where you can choose the questions you do, how do you approach which ones to pick? Difficulty (any ...
6
votes
1answer
67 views

Conservativity of $\mathrm{ZFC}+\varphi$, where $\varphi$ contradicts CH.

It is well-known that ZFC with the continuum hypothesis is a $Π^2_1$-conservative extension of ZFC. General question. What is known about the conservativity of $\mathrm{ZFC}+\varphi$ over ...
2
votes
2answers
89 views

Second Course in Number Theory - Self Study

I just finished a first course in number theory using Dudley's Elementary Number Theory. This was by far my favorite math course and I want to learn more number theory this summer. As far as ...
5
votes
2answers
198 views

How to think of zeros of the derivative of a holomorphic funcion?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, then, for example, if $f'(x_0)=0$ and $f'$ is again differentiable at $x_0$ and $f''(x_0)\neq 0$, then $f$ has a maximum or minimum at $x_0$. ...
13
votes
2answers
868 views

Does a mathematician study more, or research more?

Lately I've been studying graduate mathematics courses really hard day and night. Realizing that the things I'm learning (currently basic manifold theory and commutative algebra) are really an epsilon ...
3
votes
3answers
95 views

Colloquialisms in Math Terminology

What are some of your favorite colloquial sounding names for mathematical objects, proofs, and so on? For example, manifolds are often described using an atlas and a neighborhood describes a small ...
2
votes
2answers
239 views

Why is abstract algebra so important?

In my studies of physics and mathematics, I have encountered a fair bit of geometry, Lie group and representation theory, and real and complex analysis and I understand why these branches of ...
10
votes
0answers
126 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
23
votes
6answers
1k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
1
vote
1answer
32 views

What is the remainder useful for when dividing a polynomial?

I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems. I understand the factor theorem and its application: it helps me find roots of a ...
2
votes
2answers
81 views

Lebesgue Integral, Riemann Integral and Integrals of all sorts

I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I ...
5
votes
4answers
134 views

What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?

It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ ...
1
vote
2answers
83 views

Questions about the field scientific computing

I have heard about the field of Applied and Computational Mathematics, Scientific Computing and want to get some information. Is this a combination of computer science and mathematics? What subjects ...
2
votes
4answers
193 views

What is the most fundamental trigonometric function: cosine or sine? [closed]

$$\cos(\theta) = \sin \left(\tfrac{\pi}{2} - \theta\right)$$ $$\sin(\theta) = \cos \left(\tfrac{\pi}{2} - \theta\right)$$ Both are the same entity. But is sine the copy of cosine, or is cosine the ...
3
votes
2answers
137 views

Mathematical recommendations [closed]

I was just wondering if anyone with some experience could recommend a book for one still in the early stages of their mathematical studies(first year). Maybe something related to Algebra or history of ...
38
votes
18answers
4k views

Nuking the Mosquito — ridiculously complicated ways to achieve very simple results [closed]

Here is a toned down example of what I'm looking for: Integration by solving for the unknown integral of $f(x)=x$: $$\int x \, dx=x^2-\int x \, dx$$ $$2\int x \, dx=x^2$$ $$\int x \, ...
2
votes
2answers
34 views

A result that follows from Stokes' theorem— Important?

From Stokes theorem, it is easy to prove the folowing proposition: $\int\int_\vec{S}\vec{F}d\vec{S}=0$ if $\vec{F}=curl$ $\vec{G}$ for vector fields $\vec{F}$ and $\vec{G}$ and a closed parametrized ...
4
votes
1answer
66 views

How do topologists count infinite dimensional holes?

For example, it seems like there "should" be an infinite dimensional hole (or perhaps many) in $S^1 \times S^1 \times \ldots$. (Or perhaps none...) Is there an invariant that would count it? What ...
9
votes
1answer
139 views

A graph of all of mathematics

In mathematics, one often makes (proves) statements on the basis of: Previously proven statements Axioms I like to think of these dependencies as a directed graph, with edges from the accepted ...
49
votes
16answers
5k views

Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
9
votes
7answers
416 views

Beautiful Indefinite Integrals. [closed]

These are some of the integrals with beautiful solutions I came across- $$\int \frac{x^2}{(x\sin x+\cos x)^2} dx$$ $$\int\frac {1}{\sin^3x+\cos^3x} dx$$ $$\int \frac{1}{x^4+1}dx$$ I'd love if you ...
2
votes
0answers
29 views

Lattice with $3$ operations.

If $R$ is a commutative ring and $\mathcal I(R)$ denotes its set of ideals then I know that $\mathcal I(R)$ can be looked at as a complete lattice with intersection $I\cap J$ and addition $I+J$ as ...
2
votes
0answers
48 views

Idea behind the concept of schemes

Having taken an introductory course on algebraic geometry (without introducing schemes), the notion of schemes seems to be quite unrelated to all we've done there. What are the most important reasons ...
4
votes
1answer
69 views

Possibility of publishing

First little background. I have master degree in mathematics. Then I decided to continue to study PhD level. After some years I cancel study (reason was in some things in my life). Now I am returning ...
1
vote
1answer
36 views

Prerequisites for studying Introduction to probability theory by William Feller,vol 1 & vol 2?

I know calculus, real analysis, discrete mathematics and applied probability, and want to know what else do I need to know to self-study both the probability books by Feller?
0
votes
3answers
186 views

I have a proof of the Collatz conjecture . Where should I submit my proof?

I have been working on the Collatz conjecture starting around 10 years ago. I would like to submit my proof for analysis. I would like advice on where to submit my proof.
2
votes
1answer
114 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
3
votes
0answers
68 views

Am I missing out by not knowing another language?

A bunch of famous mathematicians, e.g. Kolmogorov, `Bourbaki,' Laplace, Lebesgue etc. wrote in foreign languages and I have seen peripherally that lots of new results are published in French. ...
3
votes
0answers
116 views

History of “Math is an Art” [closed]

For all its elegance I cannot bring myself to the conclusion that math is a form of art. As shown on the wikipedia page there is certainly math in art and art in math but what I wonder is how the ...
-2
votes
1answer
142 views

How many hours is needed to finish reading a textbook in mathematics? [closed]

I know that this question does not admit a definitive answer. Surely it depends on how long and how difficult the book is, the reader's ability and background knowledge, etc. But I still want to ask ...
28
votes
27answers
2k views

Gift advice: present for high school graduate interested in math

I am a PhD student in mathematics who recently found out that I will be attending my girlfriend's cousin's high school graduation party. I have never met the cousin, but hear that he is very ...
1
vote
0answers
36 views

Choosing referees for peer review

When submitting to a journal for publication and suggesting referees expert in the field for peer-review, is it the done thing (or just polite) to contact the referees to ask if they would be happy to ...
2
votes
2answers
72 views

Has this problem been studied?

Today in Italy all students under 18 faced with a test used to establish the quality of the schools they are in. I read one of the question (a mathematical question, of course!) that make me thing for ...
3
votes
0answers
57 views

How to practice applied mathematics calculation skill

As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students ...
2
votes
3answers
94 views

Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction.

Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction. I've been thinking how to prove the above statement not using contradiction. My idea for a ...
3
votes
0answers
44 views

Efficient software for producing (expression) trees “on the fly,” with good editing (e.g. cut/copy/paste) facilities?

A formula like $\forall x \exists y(x+y=0)$ can be represented as a tree. Something like so: ...
4
votes
0answers
134 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
1
vote
0answers
101 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
68
votes
10answers
9k views

Why can't you add apples and oranges, but you can multiply and divide them?

What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided? This looks ...
8
votes
3answers
1k views

What Maths are the most important for Artificial Intelligence?

I am just curious about this. Please don't include anything about programming.
3
votes
1answer
178 views

Ultimate GRE Prep

I'm planning on taking the math GRE Subject Exam in April (~11 months from today). I want to start preparing now in the hopes of scoring in the 95+ percentile. I have already taken a number of ...
-1
votes
3answers
103 views

College Math Competitions

Next year i'll be going to UMCP and i'be been doing competition math from middle school and throughout high school. I've done some looking around, and other than the Putnam Competition, there don't ...
3
votes
1answer
47 views

Good example that enumerating notation of sets is not unique

Obviously the enumerating notation of sets like $\{ 1,2,3,4,5,\ldots \}$ is not unique because it is not clearly defined how to continue the dots "$\ldots$". For example the above defined set could ...