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18
votes
6answers
2k views

Submit papers: arxiv or vixra? [closed]

If I submit a paper at other place (for example vixra) first, can I (modify it and) submit it to arxiv again? Is it valuable to publish a paper at the vixra? Except arxiv and vixra, does any other ...
2
votes
1answer
83 views

Why is Cauchy's integral formula always written with the function as the subject? [closed]

Why is Cauchy's integral formula always written as $$f(w)=\frac1{2\pi i}\int_L\frac{f(z)}{z-w}dz$$ instead of as $$\int_L\frac{f(z)}{z-w}dz=2\pi i f(w)$$? Isn't the latter form how it's typically ...
5
votes
2answers
288 views

Going back to the basics? [closed]

I'm currently in my second year of college majoring in comp sci and I haven't really taken any math courses yet except pre-calc. In high school, I thought of myself as a pretty good math student and ...
4
votes
17answers
3k views

Can we get just $3$ from $\pi$? [closed]

Today, a friend and I solved a question and one point came up where we were discussing whether we should write $(-1)^{n-1}$ or $(-1)^{n+1}$ and quickly we remembered that it was the same thing (for ...
3
votes
1answer
134 views

Name for three-valued sign $+, -, 0$

Is there an accepted term (an adjective or prefix) like strict, trichotomous, strong or definite sign to indicate the three-valued sign whose values are $+$, $-$, and $0$? Are there words reserved ...
34
votes
6answers
2k views

A Question about Doctoral Theses in Mathematics [closed]

This is most definitely a soft question, which I'm sure may get some negative attention, and perhaps even be voted closed. However, I genuinely would like to generate answers on this matter as it ...
6
votes
4answers
252 views

What is the relationship between “recursive” or “recursively enumerable” sets and the concept of recursion?

I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am ...
3
votes
2answers
247 views

Why demonstrations are important in mathematics? [closed]

Good evening, I'm studying math and would like to know how important are mathematical proofs in the world and particularly in a school of mathematics Thanks for your help
43
votes
16answers
5k views

What is $-i$ exactly?

We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything? Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen ...
1
vote
1answer
200 views

Which topics of real-analysis should be studied if you have already done calculus

Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go ...
37
votes
3answers
2k views

Why learning modern algebraic geometry is so complicated?

Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the ...
16
votes
6answers
419 views

Why is boundary information so significant? — Stokes's theorem

Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
2
votes
1answer
65 views

About the type of numbers allowed by axioms and Nature

I have a question which has 4 different subcases or "avatars": 1) Has every "interesting" class of number been invented? 2) Has every "possible" class of number been invented? 3) Does Nature use ...
7
votes
6answers
714 views

Which topics of mathematics should I study? [closed]

I'm a first year econometrics student with a great interest in mathematics. I very much enjoy my study, but still I am interested to learn about more topics in mathematics which are not part of my ...
13
votes
3answers
242 views

Working with subsets, as opposed to elements.

Especially in algebraic contexts, we can often work with subsets, as opposed to elements. For instance, in a ring we can define $$A+B = \{a+b\mid a \in A, b \in B\},\quad -A = \{-a\mid a \in A\}$$ ...
4
votes
0answers
222 views

Soft question: why are there non-smooth manifolds?

Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...
4
votes
1answer
234 views

Getting better at math? [duplicate]

I really want to get better at math, but I'm not sure how. I have a book with competition problems from which I could learn, but I really don't enjoy sitting there for half an hour trying to find some ...
2
votes
0answers
59 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
6
votes
3answers
751 views

Importance of Neatness / Organization / Speed in Math?

Pretty simple question here but it does relate to math. I ask this as my writing is quite messy, possibly a cause of silly mistakes. How important is neatness in math? Does having messy writing put ...
7
votes
3answers
410 views

Rationale behind truth values

I originaly asked a question on Programmers.SE to know why $0$ was consider $\text{false}$ and all the other [integral] values were considered $\text{true}$. That was a huge debate and many said it ...
4
votes
0answers
184 views

Good examples of proofs in mathematics exemplary of creative reasoning [closed]

Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
8
votes
2answers
263 views

How to start learning knot theory?

Knot theory really sounds cool and I'm very interested in it. But I'm wondering what basic knowledge it is required and how I should start learning about it. Thanks
37
votes
3answers
655 views

Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
6
votes
7answers
1k views

Guides/tutorials to learn abstract algebra?

I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not ...
1
vote
0answers
100 views

How should we study maths? [duplicate]

what is the best way to study maths ? also , should we memorize the proofs of the theorems ? or just read them and understand them but not memorize ? also , should we follow one text or more than ...
7
votes
2answers
256 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
3
votes
1answer
907 views

Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
1
vote
1answer
72 views

Special numbers in patterns and the reasons they are special

I know there are several big list questions out there (e.g. Patterns that break down at certain numbers) that touch on classifications of mathematical structures where certain numbers don't fit in, ...
3
votes
1answer
58 views

Intended Audience for N Is a Number?

Is the film N is a Number appropriate (mathematically) for a student entering high school?
11
votes
1answer
355 views

List videos of interesting courses at the doctoral level.

Many mathematics departments has provided video lessons their courses (usually one semester) that are offered in their doctoral programs in mathematics. Most often these courses total average of 26 ...
9
votes
4answers
294 views

Should $\mathbb{N}$ contain $0$? [closed]

This is a classical question, that has led to many a heated argument: Should the symbol $\mathbb{N}$ stand for $0,1,2,3,\dots$ or $1,2,3,\dots$? It is immediately obvious that the question is ...
3
votes
4answers
272 views

PDEs in biology

I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it ...
5
votes
3answers
240 views

Statistics Workshop for High School Students

We are going to hold an introductory workshop about the statistics. The participants will be students who have just finished their 8th or 9th grade. The workshop consists of 10 two-hour sessions. The ...
2
votes
0answers
52 views

Request for translation from Russian: 'Bayesian Sufficiency' from a paper by Kolmogorov

The following seminal paper by the great Kolmogorov introduced the important statistical concept of Bayesian Sufficiency. Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi ...
15
votes
5answers
1k views

Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
22
votes
4answers
645 views

Why do we study prime ideals?

I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
3
votes
0answers
163 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
2
votes
2answers
39 views

Repeating u-substitution

A question about the integration technique of u-substitution: Is it allowed to apply u-substitution over and over again, to reduce the integral to a more manageable form?
1
vote
0answers
54 views

Improving my Mathematics [duplicate]

This isn't a regular maths question. So I'm currently in my first year at Uni doing a Mathematics major. I would like to know what things I can do to help me improve in Mathematics? I would say I am ...
14
votes
2answers
434 views

Math blogs, pros and cons for writers?

I regularly read blogs by three mathematicians, and occasionally run into others. Definitely they help me a lot studying mathematics. But now I am more interested in the writers' perspective, and I ...
1
vote
2answers
120 views

Why do we name equations more in applied math?

In pure math, equations aren't often named. For instance, we might define that $f$ is a (real) polynomial function iff there exists a finite sequence of real numbers $a_i$ such that $$f(x)=a_0 + a_1 x ...
12
votes
6answers
365 views

Should every group be a monoid, or should no group be a monoid?

Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint? Additional discussion. Define a monoid as follows. Defn 1. A ...
3
votes
1answer
134 views

Can we use proper classes in this way, to define a new infinity larger than |Ord|?

I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...
8
votes
4answers
506 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
45
votes
2answers
1k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
23
votes
5answers
460 views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
40
votes
3answers
967 views
25
votes
4answers
1k views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
2
votes
1answer
43 views

The best softwares to understand the intersections of the 3D objects in the Euclidean space

What is the best software (Easy to follow and clear graphics) to draw the intersections between two spheres, Two spheres and a pyramid, for example. The centre and the radius of the spheres are given ...
2
votes
0answers
32 views

Expectation of the area [duplicate]

Choose randomly three points in the unit square $D=\{(x,y)\mid 0\leq x, y\leq 1\}$, is it possible to calculate the expectation of the area of the triangle with the three points as vertexes? (Of ...