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1500 questions
119
votes
13 answers
In calculus, which questions can the naive ask that the learned cannot answer?
Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them. Calculus is not known to be such a field, as far as I know. (For…
119
votes
18 answers
A good way to retain mathematical understanding?
What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of core classes and easy math classes because there are…
010110111
- 2,231
119
votes
12 answers
Why are mathematical proofs that rely on computers controversial?
There are many theorems in mathematics that have been proved with the assistance of computers, take the famous four color theorem for example. Such proofs are often controversial among some mathematicians. Why is it so? I my opinion, shifting from…
Gerard
- 4,396
119
votes
5 answers
Pointwise vs. Uniform Convergence
This is a pretty basic question. I just don't understand the definition of uniform convergence. Here are my given definitions for pointwise and uniform convergence: Pointwise convergence: Let $X$ be a set, and let $F$ be the real or complex numbers.…
Jeff
- 1,751
119
votes
15 answers
How to prove that exponential grows faster than polynomial?
In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$ , $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to prove this theorem.
LeafGlowPath
- 7,723
119
votes
6 answers
The Langlands program for beginners
Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program and its geometric analogue? What are the good books…
ABC
- 2,033
119
votes
9 answers
Why should I "believe in" weak solutions to PDEs?
This is a sort of soft-question to which I can't find any satisfactory answer. At heart, I feel I have some need for a robust and well-motivated formalism in mathematics, and my work in geometry requires me to learn some analysis, and so I am…
A. Thomas Yerger
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119
votes
1 answer
$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?
If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here . Two weeks and…
Lincoln Blackham
- 1,191
119
votes
1 answer
Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$
Is there a way to assess the convergence of the following series? $$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$ From numerical estimations it seems to be convergent but I don't know how to prove it.
Leonardo Massai
- 1,943
119
votes
4 answers
Element-wise (or pointwise) operations notation?
Is there a notation for element-wise (or pointwise) operations? For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same length z, where $z_i = x_i * y_i$ . In mathematical…
levesque
- 1,469
119
votes
8 answers
Is the vector cross product only defined for 3D?
Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and $\vec b$,…
VF1
- 2,073
119
votes
13 answers
Does commutativity imply Associativity?
Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples? NOTE: I wasn't sure how to tag this so feel free…
Eugene
- 7,832
119
votes
21 answers
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives , by Leonard Mlodinow, Vintage Books, 2008. On p.107…
NotSuper
- 1,883
119
votes
13 answers
Why, intuitively, is the order reversed when taking the transpose of the product?
It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting on socks and shoes is taking the shoes off, followed…
user1337
- 25,029
118
votes
5 answers
What is the difference between a Hamel basis and a Schauder basis?
Let $V$ be a vector space with infinite dimensions. A Hamel basis for $V$ is an ordered set of linearly independent vectors $\{ v_i \ | \ i \in I\}$ such that any $v \in V$ can be expressed as a finite linear combination of the $v_i$'s; so $\{ v_i \…
Lor
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