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1500 questions
60
votes
13 answers
Why is cross product defined in the way that it is?
$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product?
So why is cross product defined…
koe
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60
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13 answers
Why does $\frac{1}{x} < 4$ have two answers?
Solving $\frac{1}{x} < 4$ gives me $x > \frac{1}{4}$. The book however states the answer is: $x < 0$ or $x > \frac{1}{4}$.
My questions are:
Why does this inequality has two answers (preferably the intuition behind it)?
When using Wolfram Alpha it…
Augusto Dias Noronha
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60
votes
9 answers
Good examples of double induction
I'm looking for good examples where double induction is necessary. What I mean by double induction is induction on $\omega^2$. These are intended as examples in an "Automatas and Formal Languages" course.
One standard example is the following: in…
Yuval Filmus
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60
votes
7 answers
How do we know the ratio between circumference and diameter is the same for all circles?
The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same for every circle?
How do we know that $\pi =…
Happy
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60
votes
4 answers
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some serious flaws.
The most natural way to fix all the…
gifty
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60
votes
7 answers
Is there an easy way to see associativity or non-associativity from an operation's table?
Most properties of a single binary operation can be easily read of from the operation's table. For example, given
$$\begin{array}{c|ccccc}
\cdot & a & b & c & d & e\\\hline
a & e & d & b & a & c\\
b & d & c & e & b & a\\
c & b & e & a &…
celtschk
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60
votes
1 answer
Penrose's remark on impossible figures
I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the group operation is - composition of…
anon
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60
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3 answers
"Every linear mapping on a finite dimensional space is continuous"
From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector spaces (not necessarily the same), as along as the domain is…
Tim
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59
votes
1 answer
Why is the absolute value function not differentiable at $x=0$?
They say that the right and left limits do not approach the same value hence it does not satisfy the definition of derivative. But what does it mean verbally in terms of rate of change?
user187397
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59
votes
5 answers
Why divide by $2m$
I'm taking a machine learning course. The professor has a model for linear regression. Where $h_\theta$ is the hypothesis (proposed model. linear regression, in this case), $J(\theta_1)$ is the cost function, $m$ is the number of elements in the…
Daniel Node.js
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59
votes
5 answers
Are functions of independent variables also independent?
It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed.
If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$…
LLS
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59
votes
2 answers
What are the rules for equals signs with big-O and little-o?
This question is about asymptotic notation in general. For simplicity I will use examples about big-O notation for function growth as $n\to\infty$ (seen in algorithmic complexity), but the issues that arise are the same for things like $\Omega$ and…
hmakholm left over Monica
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59
votes
12 answers
Very good linear algebra book.
I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), linear maps and their matrix representation and…
Mike
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59
votes
5 answers
Subgroups of finitely generated groups are not necessarily finitely generated
I was wondering this today, and my algebra professor didn't know the answer.
Are subgroups of finitely generated groups also finitely generated?
I suppose it is necessarily true for finitely generated abelian groups, but is it true in…
crasic
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59
votes
3 answers
Areas versus volumes of revolution: why does the area require approximation by a cone?
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. However, if we want to find the surface area,…
Eric O. Korman
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