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1500 questions
75
votes
3 answers
What's an example of a discontinuous linear functional from $\ell^2$ to $\mathbb{R}$?
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$…
FPP
- 2,183
75
votes
9 answers
Best Algebraic Topology book/Alternative to Allen Hatcher free book?
Allen Hatcher seems impossible and this is set as the course text?
So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.
Any good intro to Algebraic topology books?
I can…
simplicity
- 3,804
75
votes
2 answers
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
In my answer here I prove, using generating functions, a statement equivalent to
$$\sum_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$$
when $n$ is even. (Clearly the sum is $0$ when $n$ is odd.) The nice expression on the…
Mike Spivey
- 56,818
75
votes
4 answers
Check if a point is within an ellipse
I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.
How do I determine if a point $(x,y)$ is within the area bounded by the ellipse?
75
votes
2 answers
Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?
I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD.
I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$…
Danielle Intal
- 1,372
75
votes
5 answers
What does "curly (curved) less than" sign $\succcurlyeq$ mean?
I am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)
$$f(x^*) \succcurlyeq \alpha$$
and curved less than or equal to (\preccurlyeq)
$$f(x^*) \preccurlyeq \alpha$$
Can someone explain…
Dinesh K.
- 851
75
votes
2 answers
What is the difference between projected gradient descent and ordinary gradient descent?
I just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of projected gradient descent? What does that projection…
erogol
- 1,177
75
votes
4 answers
How to calculate $\,(a-b)\bmod n\,$ and $ {-}b \bmod n$
Consider the following expression:
(a - b) mod N
Which of the following is equivalent to the above expression?
1) ((a mod N) + (-b mod N)) mod N
2) ((a mod N) - (b mod N)) mod N
Also, how is (-b mod N) calculated, i.e., how is the mod of a…
J.P.
- 915
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75
votes
4 answers
Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions
Let $f_n(x)$ be recursively defined as
$$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$
i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$:
$$f_1(x)=\sqrt{x+1},\ \ \ f_2(x)=\sqrt{x+\sqrt{x+1}},\ \ \…
Vladimir Reshetnikov
- 32,650
75
votes
12 answers
Why is empty set an open set?
I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition.
The definition of an open set: a set S in n-dimensional space is…
python3
- 3,660
75
votes
1 answer
Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral:
$$\mathcal{I}(\mu,\nu)=\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$
where $J_\mu(x)$ is the Bessel function of the first kind:
$$J_\mu(x)=\sum…
Vladimir Reshetnikov
- 32,650
75
votes
1 answer
About Euclid's Elements and modern video games
Update (6/19/2014) $\;$ Just wanted to say that this idea that I posted more than a year ago, has now become reality at: http://euclidthegame.com/
12.292 users have played it in 96 different countries, and 1232 people have reached level 20 :)
Update…
Kasper
- 13,940
75
votes
2 answers
What is the difference between the Jacobian, Hessian and the Gradient?
I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me.
From my understanding, The gradient is the slope of the most rapid descent. Modifying your position…
Pluviophile
- 1,001
75
votes
2 answers
Order of general- and special linear groups over finite fields.
Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have?
$\text{GL}_n(\mathbb{F}_3)$
$\text{SL}_n(\mathbb{F}_3)$
Here GL is the general linear group, the group of invertible n×n matrices,…
user9656
75
votes
7 answers
Proof that Pi is constant (the same for all circles), without using limits
Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?
Chris Card
- 2,108