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27
votes
7answers
3k views

Why does topology rarely come up outside of topology?

I am currently taking topology and it seems like a completely different branch of math than anything else I have encountered previously. I find it a little strange that things are not defined more ...
0
votes
0answers
34 views

Will this method consider as an advantage or even more troublesome?

In my previous post, 3Dimensional runge kutta and Euler method ( help to verify the idea and proposition), the explanation may be long and unclear. Here i just demonstrate an example and would like to ...
1
vote
0answers
33 views

Looking for an “irreducible equivalence predicate” on $\mathbb{Z}$

NB: although I find it easier to couch the question below in somewhat abstract mathematical terms, I am ultimately interested in functions and data types that I can easily and efficiently implement in ...
5
votes
3answers
203 views

How to determine if I'm talented enough to study math? [closed]

After getting Bachelor's degree from Computer Science I changed my field to Applied Mathematics. The previous degree was mostly programming-oriented, so I know quite a lot about software engineering, ...
2
votes
5answers
144 views

Assertion not well-defined

My question is the following : does the assertion $$x=2\implies x^2=4 \tag1$$ have sense without declaring what $x$ should be, or should I write something like $$\forall x\in\mathbb{R},x=2\implies ...
0
votes
0answers
21 views

In mathematical writing, what does the phase “Class of nonlinear system” mean?

In many engineering and applied mathematics resources, author specify his work or theory for "a class of Y , Y={Nonlinear systems, Hybrid systems, discontinuous systems .. etc}" what does this mean? ...
4
votes
1answer
87 views

How to get some skills to solve Topology Problems? [closed]

I have started reading point-set topology from Topology-J.R.Munkres. I have read the chapter Countability and Separation Axioms and started doing the exercises on Article 30. I feel that I have ...
47
votes
17answers
2k views

What are some math books written in dialogue or story form, e.g., a teacher explaining to a student?

Good examples would be The Square Root of 2 by David Flannery or Math Girls by Hiroshi Yuki.
11
votes
0answers
94 views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
-2
votes
1answer
56 views

Math game for the kids [closed]

On my spare time I am going to program a computer math game for the kids and their friends. I'm thinking about using plenty of mathematical animations, pictures, fun facts, stories etc. I want to ...
3
votes
3answers
88 views

Is there any way to universally define the notion of $\text{Isomorphism}$?

Suppose we want to give a very general definition of the term Isomorphism, first of all, we'll want an isomorphism to be a bijective function. Informally, we want our function to preserve whatever ...
7
votes
1answer
117 views

Serendipitous mathematical discoveries in recent times

As of today, most important results in mathematics are conjectured long before they are proven. Are there any examples of (important) mathematical discoveries that were proven by chance rather than ...
98
votes
19answers
4k views

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved ...
0
votes
1answer
24 views

Best strategy to detect if 2 words are different while uveiling as little as possible

I though of the following problem; I did not have time to seriously think about and I probably won't, but rather than forgetting it I thought, maybe, someone will like it so I decided to post it here. ...
1
vote
0answers
20 views

Regarding the numbering of $n$-categories.

(By an $n$-category, I mean an $(n,n)$-category.) This is potentially a dumb question, but its been on my mind for a couple of years now, so I'm throwing in the towel and asking it. If it is a dumb ...
0
votes
1answer
35 views

A box comprised of infinite number of small similar boxes.

On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions" I don't understand what does Wikipedia wants to say with a box containing ...
4
votes
0answers
77 views

Mathematics Wallpapers

I know that this sounds very silly. But I don't know where else to ask. Is there a good free site for mathematics wallpapers , pictures etc ? Most of the time it is very difficult to find exact ...
3
votes
4answers
126 views

Help Self-Studying Calculus

I wanted to learn calculus but I have been told that you can't learn it without first learning elementary algebra. Can someone help me devise a plan for self-study because I don't know were to begin.
6
votes
0answers
52 views

Stiff Nonlinear Differential Equations

As far as I know, the concept of stiffness is hard to define rigorously, but there are plenty of handwavy descriptions and motivating examples in the literature when it comes to linear differential ...
1
vote
2answers
64 views

Which of the Following Sets are compact (C.S.I.R 2015)

$\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 1 \}$ in the Euclidean Topology $\{ (z_1,z_2,z_3) \in \mathbb C^3 : z_1^2 + z_2^2 + z_3^2 = 1 \}$ in the Euclidean Topology. $\prod_{n=1}^{\infty} A_n$ ...
3
votes
0answers
35 views

What are some concrete examples of how Graph Theory can be applied to Finite State Machines?

Conceptually, Graph Theory is not very hard to understand, even its terminology is very intuitive, however, it appears to have a very wide range of applications, one of which is the field of Finite ...
0
votes
1answer
108 views

Questions wrong or theorems useless because hypothesis impossible [closed]

Consider the following question: The area of a right triangle with integral sides is 5. What is twice its area? The answer is 10. But on closer inspection, there is no such right triangle. What can ...
1
vote
1answer
46 views

Free software (online is better) to divide large numbers?

This may be a strange request, but I need to divide the number $$\begin{align} ...
3
votes
1answer
60 views

Reading proofs vs. Attempting proofs, which one is more helpful?

In my discrete math course, I am often finding myself spending way too much time attempting a single math proof. I am starting to think that reading as many proof solutions as possible is a better ...
13
votes
0answers
166 views

Errors in math research papers

Have there been cases of errors in math papers, that were undetected for so long, that they caused subsequent errors in research, citing those papers. ie: errors getting propagated along. My ...
0
votes
1answer
54 views

List of Indeterminate forms in Mathematics

I know that , $1) \frac{0}{0}$ $2) \frac{\pm\infty}{\pm\infty}$ $4) 0 \times(\pm\infty) $ are Indeterminate forms. But in measure theory $ 0 \times(\pm\infty) =0 $ Are there any other ...
10
votes
2answers
98 views

General audience books that teach deep mathematics

When I was younger some of my favorite books were Raymond Smullyan's puzzle books. He strove to write books that covered topics more deep than more standard children's puzzles; To Mock a Mockingbird ...
3
votes
3answers
48 views

Is there a platform I can use to do my Maths assignment?

If one wants to do Maths assignment on the computer, how would he/she go about doing it? I want to be able to use all the mathematical symbols, matrices, formulas and whatnot- the same way ...
0
votes
0answers
43 views

What are some motivations that led to the development of the Lebesgue integral?

There are many kinds of integrals, with the most famous being the Riemann integral which is taught in elementary calculus classes. The motivation behind the Riemann integral is to find the area ...
5
votes
4answers
228 views

How fundamental is Euler's identity, really?

Euler's identity, obviously, states that $e^{i \pi} = -1$, deriving from the fact that $e^{ix} = \cos(x) + i \sin(x)$. The trouble I'm having is that that second equation seems to be more of a ...
3
votes
0answers
46 views

Meaning of / intuition for contraction of tensors (in the Riemannian setting)

I'm currently taking a course in differential geometry, and we are, I'm guessing, finally going to start working with the Riemannian curvature tensor after having covered a lot of smooth manifold ...
1
vote
0answers
34 views

Is it possible to visit all points on a differentiable function by “rolling”?

I recall from a discussion thread some week ago we talked about different ways to pedagogically explain differentiability. So I came up with this idea that if there for each point there exists a ...
-1
votes
1answer
45 views

As a 1st year undergrad, is it worth investing time on going through some of the IIT JEE calculus materials? [closed]

I am currently a 1st year undergrad in UK studying mathematics with interests in entering the academia. Yesterday when I was looking for some problems in analysis, I came across with this IIT JEE ...
1
vote
0answers
16 views

average order of basic trigonometric functions

In connection to a problem in a Fourier Analysis book, I had to calculate the average order of some functions. Then I came across this very elementary yet very interesting observations which follow: ...
45
votes
9answers
1k views

Examples of group-theoretic results more easily obtained through topology or geometry

Earlier, I was looking at a question here about the abelianization of a certain group $X$. Since $X$ was the fundamental group of a closed surface $\Sigma$, it was easy to compute $X^{ab}$ as ...
1
vote
1answer
24 views

Number of way that set of point can be colinear

Assume I have $n$ points in a plane. and I want arrange them in the way that for any point at least I can find two other points that are all the three points are collinar. I want to know how many way ...
6
votes
1answer
164 views

Description of the Universe $V$ [closed]

For me, the concept "set" seams very ambiguous. This does not satisfy me because sets are used very often in mathematics, and so many questions in mathematics are not definite for me. I want to read ...
2
votes
1answer
46 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
9
votes
5answers
493 views

Formulae of the Year $2016$ [closed]

Soon it's the year $2016$. Time to ponder how we can arrange the digits in 2016 to form a valid equation. Use any symbols you like (please explain the less obvious ones). Keep digits in the same order ...
3
votes
2answers
58 views

Does a random walk with infinite mean ever converge to anything?

Suppose we have a random walk on the real line whose step sizes have finite variance. We know that, when viewed as a function and suitably rescaled, this random walk will converge to a Brownian ...
4
votes
1answer
36 views

What is the minimum number of positive integer values at which uniquely determines a polynomial with integer co-efficients?

Let $p(x)$ be a polynomial with positive integer co-efficients; then what is the minimum number $k$ (depending on $p(x)$) of distinct positive integer(s) $r_1,..,r_k$ such that knowing ...
3
votes
3answers
106 views

What to teach first: Riemann sums or anti-derivative? [closed]

In some text books, I see that they teach Riemann sums first. In other texts, I see they teach anti-derivatives first. Is there any pedagogical preference? It seems to me that we should teach ...
0
votes
2answers
57 views

Are Standalone Introductory Linear Algebra Classes Bad? [closed]

The author of the textbook I will be using for my combination linear algebra/differential equations course next semester, Introduction to Differential Equations by Michael E. Taylor ...
1
vote
1answer
39 views

What is the relationship between the unit simplex and the nonnegative orthant?

I came upon this figure while reading something online. Pictured above is the intersection of the unit sphere in the nonnegative orthant and the unit simplex. Question: What is the relationship ...
5
votes
1answer
593 views

What are some major open problems in Galois theory?

Few days back one of my friend and I were discussing about Galois life and his ideas. Though we are not trained in Galois theory, but I am recently started to learn it by myself and hope to take up ...
8
votes
2answers
499 views

I like math, but can't keep up with the pace. [closed]

I'm a math major. I like math. I'm comfortable with it. I'm considering to do a PhD. The thing is I always fall behind. I can't keep up with the professor, can't turn in satisfactory homework in ...
23
votes
4answers
1k views

What's so special about characteristic 2?

I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of ...
10
votes
1answer
156 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
0
votes
1answer
40 views

Catalan Numbers vs Bell Numbers

I did a lot of research on Catalan Numbers and I came across one interesting fact that the nth Catalan numbers never exceeds the nth Bell number. I know that the nth bell numbers counts the number of ...
0
votes
1answer
34 views

Techniques of division by numbers in base n

Our current number system is in base 10, so we have devised techniques when a number is divided by a power of 10. For example: $\dfrac{350}{100} = 3.5$, by moving the decimal by two places because 100 ...