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1500 questions
68
votes
1 answer
Properly discontinuous action: equivalent definitions
Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is…
Pedro
- 6,518
68
votes
1 answer
Semi-direct v.s. Direct products
What is the difference between a direct product and a semi-direct product in group theory?
Based on what I can find, difference seems only to be the nature of the groups involved, where a direct product can involve any two groups and the…
retro
- 791
67
votes
2 answers
Differentiating an Inner Product
If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components shows that
$$
\frac{d}{dt} \langle f, g \rangle…
ItsNotObvious
- 10,883
67
votes
4 answers
Why we consider log likelihood instead of Likelihood in Gaussian Distribution
I am reading Gaussian Distribution from a machine learning book. It states that -
We shall determine values for the unknown parameters $\mu$ and
$\sigma^2$ in the Gaussian by maximizing the likelihood function. In practice, it is more convenient…
Kaidul Islam
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- 1
- 6
- 6
67
votes
9 answers
When to give up on a hard math problem?
I practice olympiad problems from books like Putnam and Beyond. Often I come across a problem that I simply can't solve. After $\sim30$ minutes of deep thinking it feels like I'm ramming my head into a brick wall, since I've exhausted all avenues of…
user1299784
- 2,009
67
votes
6 answers
Origin of the dot and cross product?
Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product…
Curiosity
- 1,568
67
votes
3 answers
The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$.
Let me try to elucidate my understanding of the…
Harry Williams
- 1,779
67
votes
4 answers
Finite Groups with exactly $n$ conjugacy classes $(n=2,3,...)$
I am looking to classify (up to isomorphism) those finite groups $G$ with exactly 2 conjugacy classes.
If $G$ is abelian, then each element forms its own conjugacy class, so only the cyclic group of order 2 works here.
If $G$ is not abelian, I am…
RHP
- 2,553
67
votes
1 answer
How to prove this recurrence relation for generalized "rounding up to $\pi$"?
The webpage Rounding Up To $\pi$ defines a certain "rounding up" function by an extremely simple procedure:
Beginning with any positive integer $n$, round up to the nearest multiple of $n-1$, then up to the nearest multiple of $n-2$, and so on, up…
r.e.s.
- 14,371
67
votes
28 answers
Is there a great mathematical example for a 12-year-old?
I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets.
Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and…
Mark Bennet
- 100,194
67
votes
2 answers
Number of monic irreducible polynomials of prime degree $p$ over finite fields
Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$?
Thanks!
IBS
- 4,155
67
votes
7 answers
How to find the factorial of a fraction?
From what I know, the factorial function is defined as follows:
$$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$
And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, $\frac{1}{2}!$, which they…
Cisplatin
- 4,675
- 7
- 33
- 58
67
votes
18 answers
Unsolved Problems due to Lack of Computational Power
I was recently reading up about computational power and its uses in maths particularly to find counterexamples to conjectures. I was wondering are there any current mathematical problems which we are unable to solve due to our lack of computational…
user671231
67
votes
1 answer
Why is TREE(3) so big? (Explanation for beginners)
I am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like.
I am watching a series of videos from David Metzler on YouTube. I have a basic understanding of some fast growing functions.
David…
Josh Kerr
- 847
67
votes
8 answers
Entire one-to-one functions are linear
Can we prove that every entire one-to-one function is linear?
Petey
- 671