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1500 questions
81
votes
3 answers
Mathematicians shocked(?) to find pattern in prime numbers
There is an interesting recent article "Mathematicians shocked to find pattern in "random" prime numbers" in New Scientist. (Don't you love math titles in the popular press? Compare to the source paper's Unexpected Biases in the Distribution of…
Tito Piezas III
- 60,745
81
votes
5 answers
Use of "without loss of generality"
Why do we use "without loss of generality" when writing proofs?
Is it necessary or convention? What "synonym" can be used?
Pedro
- 125,149
- 19
- 236
- 403
80
votes
3 answers
Fourier Transform of Derivative
Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$
What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f'(t) \…
80
votes
3 answers
String Theory: What to do?
This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this.
Anyways, I'm currently a 3rd year undergraduate starting to more seriously research possible grad schools. I find myself in…
Jonathan Gleason
- 8,192
80
votes
8 answers
How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?
How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$
I found the above interesting identity in the book $\bf \pi$ Unleashed.
Does anyone knows how to prove it?
Thanks.
Neves
- 5,701
80
votes
9 answers
What is mathematical research like?
I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or…
TheHopefulActuary
- 4,850
- 12
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- 81
80
votes
8 answers
Koch snowflake paradox: finite area, but infinite perimeter
The Koch snowflake has finite area, but infinite perimeter, right?
So if we make this snowflake have some thickness (like a cake or something), then it appears that you can fill it with paint like this ($\text{finite area} \times \text{thickness}…
ArtisanRtasin
- 743
80
votes
4 answers
Intuitive explanation of a definition of the Fisher information
I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this:
$$I(\theta)=E_\theta\left[-\frac{\partial^2 }{\partial \theta^2}\ln P(\theta;X)\right].$$
Could anyone…
maple
- 3,023
80
votes
4 answers
"Closed" form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.
One series that popped into our…
user17762
80
votes
4 answers
Gaussian distribution is isotropic?
I was in a seminar today and the lecturer said that the gaussian distribution is isotropic. What does it mean for a distribution to be isotropic? It seems like he is using this property for the pseudo-independence of vectors where each entry is…
Astaboom
- 987
80
votes
5 answers
What exactly is Laplace transform?
I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out that it gives "less familiar" frequency view.
My…
hasExams
- 2,341
80
votes
2 answers
Inscribing square in circle in just seven compass-and-straightedge steps
Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the circle — in just seven elementary steps. Euclidea…
PDE
- 1,477
80
votes
5 answers
What's so special about characteristic 2?
I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of such a field I could think of is…
Juan Sebastian Lozano
- 2,852
80
votes
7 answers
Functions that are their own inverse.
What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $
Later I heard about a fifth…
Willemien
- 6,730
80
votes
9 answers
Do "Parabolic Trigonometric Functions" exist?
The parametric equation
$$\begin{align*}
x(t) &= \cos t\\
y(t) &= \sin t
\end{align*}$$
traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly,
$$\begin{align*}
x(t) &= \cosh t\\
y(t) &= \sinh t
\end{align*}$$
draws the right part…
Argon
- 25,971