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6
votes
9answers
1k views

Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
2
votes
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 ...
1
vote
3answers
117 views

Definition of the mathematical proof

How do we define a mathematical proof? Is it a series of arguments? Is it a series of conclusions? Is it manipulation of formulas? Is it a mixture of laws of logic and axioms,theorems or ...
1
vote
2answers
31 views

Does Convexity play a role in topological equivalence?

Does convexity of two shapes play a role in them to be topologically equivalent? For example, a circle and a heart (as asked in another question I posted). They are homeomorphic, but the circle is ...
1
vote
1answer
393 views

Question about the mathematics in actuarial studies

I tried Google but there isn't much information on this and I would really like some insights into actuarial studies, the mathematics involved and how it compars to the mathematics in a bachelor of ...
0
votes
0answers
22 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
4answers
93 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 ...
1
vote
2answers
65 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 ...
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 ...
11
votes
6answers
147 views

Where to find interesting integrals for a Calc III student?

I apologize in advance if this is a very soft question. I won't be surprised or offended if I can't get a good answer. One of my favorite things to do in my spare time, when I'm feeling analytical of ...
4
votes
1answer
65 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 ...
2
votes
1answer
47 views

How to know what kinds of substitution can we do in math?

I have seen in many contexts that somebody out of the blue decides to put $x=y^2$, or $x=t/2$. So how do I know what kind of sustitution I'm allowed to do? Is there any necessary conditions or we ...
94
votes
24answers
21k views

List of Interesting Math Blogs

I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level ...
1
vote
1answer
78 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 ...
0
votes
1answer
42 views

A problem in understanding principal root in the complex plane.

We know that every complex number has exactly $n$ $n$-th roots in the complex plane, and we usually take (if the context where we are working doesn't tell us more) the one with real and imaginary part ...
2
votes
0answers
63 views

Cognitive processes involved solving IMO level problems [closed]

I am currently 16 years old and, though I'm obviously not as good as most of the people on this site, I have always been considerably better than most of my classmates in mathematics. This, of course, ...
6
votes
1answer
300 views

What is meant by “folkloric result” in category theory?

I often will see the word "folklore" used in papers on category theory, e.g. in Barr's paper on Isbell Duality, he states a result, which he proves, is "folkloric", on page 512: The following ...
14
votes
4answers
375 views

Which mathematical ideas most influenced the way you think?

This is not a question about how you use a formula or mathematical method to solve quantitative problems - that is applied mathematics. Rather, I'd like to hear how deeper ideas gained through the ...
8
votes
1answer
56 views

Circle and heart homeomorphic?

Is a circle and heart homeomorphic to one another? Intuitively, I can picture that the one can be "morphed" into the other by bending and stretching and not breaking. But I am unsure if that is ...
4
votes
5answers
1k views

Graph Theory Applications?

What are the areas where graph theory can be applied? Cause I wonder what applications this have on the real world. Does it solve certain problems and stuff? Areas such as communication networks ...
0
votes
0answers
45 views

Real or Complex Analysis next quarter?

I apologize if this is the wrong place for to ask a question about a math course I should take, but I didn't see a better stack-exchange forum for this. Long story short, I am a fourth-year chem PhD ...
3
votes
2answers
615 views

What's the most effective ways of teaching kids - times tables?

I'd like to help a $6$ year old who already has a pretty good grasp of $2$, $5$, and $10$ times tables.
3
votes
0answers
35 views

Criterion for Improvement

I was recently asked the question "How do you know when you've become a better mathematician/better at mathematics?," and I realized that at that moment I did not have a valid answer, since I have ...
0
votes
0answers
13 views

Good introductory texts on modular forms/L-functions

I am relatively new to these areas but would like to gain some understanding through an introductory text. I am an undergraduate math major so ideally these books should be accessible to someone with ...
5
votes
0answers
67 views

Is it worth proving every theorem you learn?

Ever since I determined mathematics - mainly set theory and number theory - was my main passion, and I began learning mathematics formally outside of the curriculum posed within secondary schools I ...
6
votes
0answers
90 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
1
vote
0answers
20 views

Prove that $c_1 \phi_1 + c_2 \phi_2$ solves the IVP

Theorem: If $\{ \phi_1, \phi_2\}$ is a fundamental set of solutions of $$x''+p(t)x'+q(t)x=0,$$ then for any initial values $x(t_0)=x_0, x'(t_0)=y_0$, there are constants $c_1$ and $c_2$ so ...
0
votes
1answer
305 views

Convergence of a sequence in trigonometric functions

Is the sequence $(a_n)$ defined as $$a_n :=\dfrac {\sin\Big( \dfrac {\pi}4+\dfrac n2 \Big)\sin\Big(\dfrac{n+1}2 \Big)}{\sin\Big(2n+2\Big)\sin\Big( \dfrac {\pi}4+\dfrac {n-1}2 \Big)\sin\Big(\dfrac{n}2 ...
2
votes
2answers
270 views

Foundations book using category theory for student embarking on PhD in mathematical biology?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
2
votes
2answers
58 views

Recommendation of multivariable calculus books

I am looking for some suggestions on a good calculus book I shall keep on hand all the time. I am a graduate student who will be commencing research in the area of theoretical PDE (nonlinear). ...
2
votes
2answers
83 views

Algebra Text Recommendations [closed]

I am looking for any recommendations or suggestions for a good book covering an introduction to the following; Relation , sets and functions, divisibility theory and modular arithmetic , groups, ...
17
votes
1answer
727 views

The most active fields of mathematics?

Which fields of mathematics are the most active at this time -- by number of papers published, grant money, people working in them or by any other measure? Any trends in this regard?
0
votes
0answers
64 views

Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation

The biggest obstacle for me to learn geometry and topology is the haziness of textbooks. I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to ...
0
votes
1answer
120 views

Latest episode of the big bang theory, vanity card.

I usually don't read these, but this time I did, and this was the card: Does the last mathematical symbols have any meaning? I get that the equal 150.6+V, is there any more meaning behind this?
6
votes
1answer
66 views

When vectors act on scalars.

Background. I've been struggling through an introduction to differential geometry this semester. Recently, a tiny part of what we've been learning "clicked" for me, and to solidify this, I'd like ...
1
vote
0answers
44 views

How to teach Mathematical Induction mathematically? [migrated]

I am exhausted of teaching Mathematical induction to my little brother. I have given him many examples, Domino effect, aligned shops of hot dogs etc and every time he says that he got it but when I ...
7
votes
6answers
115 views

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first ...
45
votes
6answers
1k views

What to answer when people ask what I do in mathematics

This is not really a math question, but I think that every mathematician and student in math (specially pure) struggles with this at some point. Inevitably at some point when we're talking with ...
2
votes
0answers
33 views

Is there a measure theoretic version of Stokes's theorem?

Is there a way to generalize Stokes's theorem on manifolds to general measure spaces? This idea came from trying to generalize the fundamental theorem of calculus to general function/infinite ...
0
votes
2answers
956 views

Formula to calculate difference between two dates

I am working on getting date difference I have a formula to calculate it as follows $$365\cdot\mathrm{year} + \frac{\mathrm{year}}4 - \frac{\mathrm{year}}{100} + \frac{\mathrm{year}}{400} + ...
6
votes
2answers
333 views

What are well-known weaknesses of CAS/math software?

I'm starting to become more functional in Mathcad, so I wanted to take the opportunity to look into known drawbacks, mathematically speaking, of CAS/computer math systems. For instance, one of the ...
6
votes
2answers
122 views

Mathematics that is not taught in college [closed]

Among the mathematics that is not usually taught in high school and college (for pure math majors), what are some: Important Interesting Elegant mathematical ideas, tools, ...
3
votes
0answers
92 views

How to study multiple books per math subject?

S.E advisers, I am a college sophomore in US with double majors in mathematics and microbiology. I apologize for this sudden interruption but I wrote this email to seek your advice regarding to ...
1
vote
2answers
37 views

SDE Modeling: Ito vs. Stratonovich

In my SDE class last semester there were some hints that sometimes an SDE model makes more sense in the Ito sense, and sometimes in the Stratonovich sense. This was explained very briefly and vaguely. ...
5
votes
2answers
82 views

What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
3
votes
1answer
65 views
46
votes
7answers
7k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
7
votes
1answer
75 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
1
vote
0answers
10 views

Good resources for learning to recognize word problems in statistics?

I've got a number of books and resources for statistics theory, but I've always had problems with the approaches needed in answering questions, specifically for probability theory where counting ...
34
votes
8answers
1k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...