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1500 questions
69
votes
6 answers
Is there a reason it is so rare we can solve differential equations?
Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complicated solutions or be unsolvable. Is there some…
novawarrior77
- 850
69
votes
4 answers
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on itself. I know that a categorical approach is…
Rachmaninoff
- 2,680
69
votes
8 answers
Good books on Math History
I'm trying to find good books on the history of mathematics, dating as far back as possible.
There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to Philosophy, and there were no good recommendations on…
Philoxopher
- 431
69
votes
8 answers
Entire one-to-one functions are linear
Can we prove that every entire one-to-one function is linear?
Petey
- 691
69
votes
3 answers
Necessity/Advantage of LU Decomposition over Gaussian Elimination
I am reading the book "Introduction to Linear Algebra" by Gilbert Strang and couldn't help wondering the advantages of LU decomposition over Gaussian Elimination!
For a system of linear equations in the form $Ax = b$, one of the methods to solve the…
69
votes
0 answers
Determinant of a matrix that contains the first $n^2$ primes.
Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix
$$
\left(\begin{matrix}
p_1 & p_2 & \cdots & p_n \\
p_{n+1} & p_{n+2} & \cdots & p_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\cdots & \cdots…
Rofl Ukulus
- 852
69
votes
7 answers
Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?
Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
Clayton Kershaw
- 693
69
votes
12 answers
Is it morally right and pedagogically right to google answers to homework?
This is a soft question that I have been struggling with lately.
My professor sets tough questions for homework (around 10 per week).
The difficulty is such that if I attempt the questions entirely on my own, I usually get stuck for over 2 hours per…
yoyostein
- 20,428
69
votes
1 answer
Baby/Papa/Mama/Big Rudin
Recently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that these are some nick names used for the books of…
Hosein Rahnama
- 15,554
69
votes
5 answers
Are there an infinite number of prime numbers where removing any number of digits leaves a prime?
Suppose for the purpose of this question that number $1$ is a prime number.
Consider the prime number $311$. If we remove one $1$ from the number we arrive at the number $31$ which is also prime. If we removed $3$ instead of $1$ we would arrived at…
Farewell
- 5,046
69
votes
6 answers
Am I too young to learn more advanced math and get a teacher?
I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics that are covered in university, just the ones…
Argon
- 25,971
69
votes
6 answers
Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only "unlikely" or that the probability approaches zero or…
regularmike
- 703
69
votes
6 answers
Is it possible to formulate category theory without set theory?
I have never understood why set theory has so many detractors, or what is gained by avoiding its use.
It is well known that the naive concept of a set as a collection of objects leads to logical paradoxes (when dealing with infinite sets) that can…
Matt Calhoun
- 4,414
69
votes
4 answers
Topological spaces admitting an averaging function
Let $M$ be a topological space. Define an averaging function as a continuous map $f:M \times M \to M$ which satisfies $f(a,b) = f(b,a)$ for all $a,b \in M$ and $f(a,a) = a$ for all $a \in M$.
These seem like reasonable properties for a function…
Steven Gubkin
- 10,018
69
votes
5 answers
Adding two polar vectors
Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?
lash
- 799