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4answers
49 views

Applying math knowledge [closed]

Currently I'm in the middle of my first year of college studying informatics engineering. I was never great at math, but if I put some effort, I understand it and constantly get good grades. However, ...
2
votes
2answers
344 views

Is Physics really a rigorous subject? [closed]

Though I can't give a precise definition of the term rigor (or better to say mathematical rigor) but intuitively in case of mathematics one may note that when we say that 'the proof is rigorous' we ...
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4answers
5k 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 + ...
3
votes
3answers
56 views

If $1$ was a prime, could it be possible for the prime factorization of any number to go on forever?

If $1$ was a prime, could it be possible for the prime factorization of any number to go on forever? I think this would happen because if you multiply anything by $1$, you get the first factor ...
6
votes
2answers
80 views

Which mathematical topics is knot theory related to?

I wonder if knot theory is related to any other topic in mathematics. I've not read much about it, but it seems to be living isolated. I also wonder if there any particular mathematical background ...
4
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3answers
81 views

Should I go back and start with a more “proof” based approach?

So I'm currently a calculus student, next semester I'll take calculus 2. I'm wondering if I should go to a book like the one by Spivak which is entirely different from the book used for my course, and ...
18
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6answers
3k views

Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
4
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2answers
73 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
16
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9answers
784 views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
156
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24answers
12k views

Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
4
votes
4answers
135 views

Transition from introduction to analysis to more advanced analysis

I am currently studying intro to analysis and learning somethings about basic topology in metric space and almost finished the course . I am thinking of taking some more advanced analysis. Would it be ...
58
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3answers
4k views

Getting Students to Not Fear Confusion

I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro ...
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6answers
3k views

Advice for Self-Study

I am a senior in high school who has taught myself through Calculus BC and I got a 5 on the exam. However, I have taken all the math I can at my school. I have also taught myself multi-variable ...
3
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1answer
73 views

Great books on all different types of integration techniques

It's coming up to Christmas so I can ask to have all the books I can't afford from begrudging relatives! I'm really interested (mainly from looking at some of the answers cleo and other fantastic ...
1
vote
1answer
49 views

What does “if and only if” mean in definitions?

Consider the following definition: A sequence $\{p_n\}$ is Cauchy if we have that for every $n, m \ge N$: $$|p_n - p_m| < \epsilon$$ Although if and only if is not used, we know that if a ...
3
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0answers
60 views

Topology of the space of “loops” [closed]

I have a question that I'm not even sure I can put into words, but please bear with me! I want to define some sort of "loop space" and I want to understand it's topology enough that I can compare it ...
4
votes
1answer
142 views

Is there any mathematician who felt guilty for one of his math discoveries ever?

Quoted from Wikipedia: In 1888 Alfred Nobel's brother Ludvig died while visiting Cannes and a French newspaper erroneously published Alfred's obituary. It condemned him for his invention of ...
2
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0answers
22 views

The idea behind the Sobolev embedding

Sobolev embedding and compact embedding are the most popular theorems in Sobolev space we actually used in research. But after I use them so many times, I am still wondering, why, philosophically, ...
7
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3answers
194 views

Is it still possible for mathematicians to contribute to the theory of music?

Is it still possible that mathematicians contribute to the theory of music? Is the mathematical foundation of music still an area of research? If yes, what new researches have been done regarding ...
1
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1answer
37 views

Studying mathematics in France (universities and grandes ecoles)

I am currently a senior in Albania and would like to study mathematics in France. However, I'm not quite sure if the universities or the so-called "grandes ecoles" provide the best quality of ...
14
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9answers
408 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 ...
9
votes
11answers
886 views

Who is our hero?

The idea of this soft question raised by a little girl when I was talking for children in a children's cancer clinic about the heaven of mathematics and its beauties during a friendly mathematical ...
11
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2answers
560 views

So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
5
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0answers
49 views

Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
9
votes
1answer
129 views

What would it mean if there was a link between e and $\pi$?

It is not even know if $\pi+e$ is rational, and the same is true for other similar expressions involving $\pi$ and $e$, but does this have an impact? If it were, for example, proven that $\pi=ae$ or ...
1
vote
1answer
81 views

Pure Mathematics vs Mathematical Statistics

I see that these majors are usually offered separately at universities. 1) Does Pure Math not cover ALL math including statistics? 2) If you choose Pure Math - will there be things you will NOT ...
8
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4answers
635 views

Generalized graph theory

This question may be kind of 'out there' but it got me thinking. In graph theory we have a set of vertices $V$ and a set of edges $E$ which is made up of 2-element subsets of $V$ (either unordered or ...
5
votes
5answers
207 views

Should i study Mathematics? [closed]

I would like to ask you guys for some advice. I'm currently studying Bsc. engineering(2nd year). I really like mathematics and for example I really liked the calculus courses from 1st year and am ...
2
votes
1answer
62 views

Getting stuck on difficult problems.

First, a little background: I hope to go to graduate school in mathematics, but for financial reasons I will be unable to go back to school any sooner than the fall of 2016. However, since I feel ...
8
votes
2answers
80 views

How to google search mathematical notions and expressions?

It is usually not difficult to google search mathematical notions; for example, one can search (with quotation marks) the term "brunnian braid" and find the definition and other related materials. ...
21
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2answers
1k views

How do people come up with difficult math Olympiad questions?

The problems that appear in difficult math competitions such as the IMO or the Putnam exam are usually very difficult and require some ingenuity to solve. They also usually don't look like they can be ...
4
votes
1answer
120 views

Is Euler's Introductio in analysin infinitorum suitable for studying analysis today?

I've read the following quote on Wanner's Analysis by Its History: ... our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than ...
8
votes
1answer
415 views

Is it possible to learn mathematics right from the source instead of reading textbooks. By studying the masters and not their pupils

I was wondering if mathematics learning process require the use of textbooks. When I was a high school student, I read as a preparation for university, Legendre book on Elements of geometry and ...
282
votes
35answers
30k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
3
votes
2answers
150 views

More than one pair of “nice” adjoint functors between different concrete categories

Though adjoint functors provide a universal description for many concrete mathematical constructions, these constructions usually revolve around finding a single "canonical" way to transform one type ...
5
votes
2answers
100 views

Still forget even if theorem-proof “self-discovered”; Importance of intuition/proficiency of concepts in research work…

It is widely said if we go through concepts/theorems/proof on our own by actively doing instead of passively reading, the idea will be ingrained in mind. I agree with that, it really often helps. ...
8
votes
10answers
350 views

Real analysis book suggestion

I am searching for a real analysis book (instead of Rudin's) which satisfies the following requirements: clear, motivated (but not chatty), clean exposition in definition-theorem-proof style; ...
0
votes
3answers
63 views

What are some textbooks on the same level as Ross's “A First Course in Probability”?

Ideally, I would like to have at least six standard probability texts so that I can compare them to each other. Thank You.
1
vote
2answers
33 views

Is there a finite vector subspace over the reals?

I cannot think of a finite vector space over the reals, because we must sum these two elements and get a new element also in the vector space. And over the reals, I can't think of a sum that, at some ...
15
votes
9answers
2k views

“Honest” introductory real analysis book

I was asked if I could suggest an "honest" introductory real analysis book, where "honest" means: with every single theorem proved (that is, no "left to the reader" or "you can easily see"); with ...
0
votes
0answers
29 views

Is there a way on MSE to search for previously asked questions using LaTeX code? [migrated]

Before I post a question, I always try to see if it has been previously posted. But for some questions with mostly "math symbols" and few words, I find it difficult to search effectively. For example, ...
16
votes
5answers
2k views

Is there a(n elementary) function whose derivative we cannot integrate?

Say, for example, I take a reasonably-complicated function $f(x)=\tanh[\ln(x^x)]$, and differentiate it to get $$f'(x)=\frac{4x^{2x} [1+\ln(x)]}{(x^{2x}+1)^2}.$$ Now, to integrate this, I imagine, ...
0
votes
1answer
19 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
2
votes
1answer
54 views

What does $S^z$ mean for each $z\in\mathbb{C}$?

Let $S$ be a set. What does $S^z$ mean for each $z\in\mathbb{C}$? In Set Theory numbers are sets and for any two sets $A$ and $B$, we define $B^A$ as the set of maps from $A$ to $B$. Well okay, ...
165
votes
85answers
14k views

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
8
votes
2answers
473 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? I'd like the lectures to cover main topics of the books ...
2
votes
1answer
80 views

Why is polynomial long division being taught in schools instead of Horner's method? [closed]

The Horner´s method is by a long shot easier than the Polynomial long division and serves the same purpose. Why isnt it being taught in school (in germany at least)?
1
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0answers
22 views

Generators of Intersection of two Subgroups

Let $G$ be a group and let $A$ be the subgroup of $G$ generated by $\{a_i\}_{i\in I}$; let $B$ be the subgroup of $G$ generated by $\{b_j\}_{j\in J}$, where $I$ and $J$ are index sets. Is there a way ...
1
vote
3answers
158 views

How much of Mathematics is limited by our writing? [closed]

I'm sorry if this question is too vague or otherwise a stupid question. Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or ...
4
votes
1answer
78 views

Is cellular automata something that is studied in mathematics departments?

I am interested in studying cellular automata but am unsure if I should be looking at CS or mathematics graduate departments. Symbolic dynamics seems to have some tie to cellular automata but I ...