Hey, @robjohn. I have a question for you - what should I do if I see a user that I think is misusing the site in a completely non-mathematical way? There's a particular user I've seen with about a half-dozen instances in the last week of serial voting reversed, and who I think is misusing the review queues.
Should I just flag a random question of this user's and explain my concerns?
Hey guys, is anyone here familiar with pseudodifferential operators? I want to show that if two $T_\alpha$ and $T_\beta$ are two elliptic pseudo-differential equations, then $T_\alpha T_\beta$ is also an elliptic pseudo-differential operator.
For any number $a \gt 0$, let $f_a$ be defined as follows:
$f_a(x)=\begin{cases}f(x-a)&x\geq a\\0&0\leq x<a\end{cases}$
Show that $L\{f_a(x) \} = \hat f(s+a)$
This is what I did so far
$f_a(x)=\begin{cases}f(x-a)&x\geq a\\0&0\leq x<a\end{cases}\\\qquad=f(x-a)h(x-a)$
$\mathcal L\{f_a(...
hi. I can answer your second question. A nontrivial subgroup is a subgroup other than {e} where e is your identity. A subgroup that has the identity element alone is a trivial subgroup which doesn't really do anything.
suppose I have a string of size $n$. How many possible permutations does it have? for example the string "NQEL" has 12 substrings: N, Q, E, L, NQ, QE, EL, LN, NQE, QEL, ELN, and LNQ
and by "substring" you mean some kind of contiguous block of letters in the original string? (for example, QL does not seem to be listed as a substring of NQEL)
and you do not count the original string as a substring, correct?
I do not think there will be any formula for this. it depends on more than just the number of letters, or even the multiplicity profile of all of the letters: it requires their exact arrangement (up to cycling). in general, you will have to settle for an algorithm to compute the number of contiguous substrings of a cyclically ordered string.
Hello All, I would like to bring your attention to this problem math.stackexchange.com/questions/575127/… I have been able to provide a partial answer. If anyone can complete the solution or provide another solution it would be nice.
The statement is incorrect. It should be that $({\cal L} f_\alpha)(s) =e^{-sa}({\cal L} f)(s)$.
You finished right before you wrote "Everything ....".
There you showed that $({\cal L} f_\alpha)(s) = \int_0^\infty f(t) e^{-s(t+a) } dt$, which is almost finished.
Continuing, $\int_0^\infty f(t) ...
why are you trying to prove an incorrect statement?
either you have misunderstood the question (and therefore miscommunicated it to us), or the problem is simply wrong, and if your instructor has any sense they should accept e.g. a proof of the (a) & (b) I gave above as remedy
go back and read what I wrote for (a) and you'll see which letter it should be! (hint: you had it correct the very line before it, but you changed the letter inexplicably)
one needs to observe that the laplace transform of $e^{-at}f(t)$ is $$\int_0^\infty e^{-at}f(t)e^{-st}dt=\int_0^\infty f(t)e^{-(s+a)t}dt=F(s+a)$$ where $F(s)=\int_0^\infty f(t)e^{-st}dt$. not hard at all.
but you hadn't even got to the proof. you were having trouble writing down the problem itself. writing down the problem is a matter of following directions. when part (b) says to apply the laplace transform to exp(-at)f(t), then that's what you do!
the laplace transform of exp(-at) is 1/(s+1), but not the laplace transform of exp(-at)f(t)
notice how 1/(s+1) does not depend on the function f(t) at all, so it should immediately ring alarm bells that 1/(s+1) is not the transform (not to mention it has nothing to do with the statement of (b))
Did any one see my question about a Dirichlet character sum for the von Mangoldt function? I would like generalize it to different values of $j$ like in this Mathematica code: Table[DirichletCharacter[7, j, n], {j, 1, EulerPhi[7]}, {n, 0, 3}] // Grid
@PabloRotondo I'm trying to find a way to elementarily compute $$\lim_{n\to\infty}\frac{1}{\sqrt{n}}\sum_{k=0}^{n}\frac{n^k \Gamma(n + 1)}{\Gamma(n + k + 1)}$$
Suppose $\gamma$ is a closed convex curve, and $u(x)=d(x,\gamma)$ is the distance function defined outside $\gamma$. Is it obvious that $\lvert\nabla u\rvert=1$?
V.I.Arnold gave $u$, then said that it's obvious when we know that the geometrical interpretation for $\lvert\nabla u\rvert$ is the maximal rate of increase.
@FrankScience You probably know that $\lvert u(x) - u(y)\rvert \leqslant \lVert x-y\rVert$, so from that $\lVert \nabla u\rVert \leqslant 1$. Now fix $x$ and consider the projection $Px$ of $x$ on the curve. Since the curve is convex, it is entirely on one side of the line orthogonal to $x-Px$ through $Px$, on the other side than $x$. So $u(x+t\cdot (x-Px)) = u(x) + t\lVert x-Px\rVert$ for $t \geqslant 0$.
That means the maximal rate of change is $1$, hence $\lVert \nabla u\rVert = 1$.
@MatsGranvik A Dirichlet character is a function $\chi:\Bbb Z/(k)^\times \to S^1$ such that $\chi(ab)=\chi(a)\chi(b)$, $\chi(1)\neq 0$ (you can extend to all of $\Bbb Z/(k)$ by $\chi(a)=0$ whenever $(a,k)>1$.)
1. I don't know. 2. I think it's challenging to know what is the probability for an $n$-dimensional point to lie on an $n$-dimensional half-space, while it follows the $n$-dimensional Gaussian distribution... No? :/
@PedroTamaroff, I forgot to mension you in my last message... Really, does it seem to you like a stupid question?
Hey people - am I right in thinking that this expression is right? mathbin.heroku.com/hPJkDUU (I'd type it into chat, but MathJax doesn't display for me in it and typing that out would be horrendous)
if anyone has a moment, I would appreciate some feedback from the types who hang out here on my project, it's a latex calculator szifler.com/calc/calculator.html
it's only a few days old so I don't expect it to be rock solid, and it would have no idea how to solve a polynomial above order 3
It really deppends on my mood, and they are hard to compare. I really liked Dark Angel when I was younger. It is not a particularly great one. Recently I really liked Black Orphan. And at the time I saw Firefly it really made an impression one me.
I think the scariest number has to be 1,000,000,000,000,066,600,000,000,000,001. It's called the Belphegor's prime. Belphegor, fyi, is a price of hell.
all I gotta do for my math is to write this down on my paper and I"m done :D. THank goodness that my calc iv homework is only 4 problems and due on Wednesday. But, I like to get things done
This is from the book Linear algebra demystified.
For any number $a \gt 0$, let $f_a$ be defined as follows:
$f_a(x)=\begin{cases}f(x-a)&x\geq a\\0&0\leq x<a\end{cases}$
Show that $L\{f_a(x) \} = \hat f(s+a)$
This is what I did so far
$f_a(x)=\begin{cases}f(x-a)&x\geq a\\0&0\leq x<a\end{cases}\\\qquad=f(x-a)h(x-a)$
$\mathcal L\{f_a(...
The statement is incorrect. It should be that $({\cal L} f_\alpha)(s) =e^{-sa}({\cal L} f)(s)$.
You finished right before you wrote "Everything ....".
There you showed that $({\cal L} f_\alpha)(s) = \int_0^\infty f(t) e^{-s(t+a) } dt$, which is almost finished.
Continuing, $\int_0^\infty f(t) ...
