Mandelbrot set is connected. That is to say within a mandelbrot set for any pair of points there is a path within the set, connecting these points. What abouthe set of all other points? Is there a pair of points that cannot be connected without going through the Mandelbrot set?
In this  of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is said that the simply connectedness of Mandelbrot set is easier to prove.  However, I cannot find a proof through google. Can anyone provide me a reference or a proof of the simply connected of Mandelbrot set?
I don't understand at all why:  \begin{equation} \sum\limits_{n=0}^\infty p_n x^n = \prod\limits_{k=1}^\infty (1-x^k)^{-1} \end{equation}  Where $p_n$ is the number of partitions of $n$. Specifically how can the sum of positive numbers ever be factored into a fraction or factors to the $-1$ power? Is this something purely notational or am I missing something big?  Thanks for all the help! For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$ The definition of a Fibonacci number is as follows:  $$F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$   Prove the given property of the Fibonacci numbers for all n greater than or equal to 1. $$F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$$  I am pretty sure I should use weak induction to solve this. My professor got me used to solving it in the following format, which I would like to use because it help me map everything out...  This is what I have so far:  Base Case: Solve for $F_0$ and $F_1$ for the following function: $F_nF_{n+1}$.  Inductive Hypothesis:  What I need to show: I need to show $F_{n+1}F_{n+1+1}$ will satisfy the given property. Proof Proper:  (didn't get to it yet)  Any intro. tips and pointers? How do I automate a key press in AppleScript? Is it possible to automate the key press of a key (such as "a" or "&lt;") in Mozilla Firefox, Google Chrome or Safari (in this order)? I'd like to automate the key press of "a" for example in Mozilla Firefox for 100 times every x seconds. With x being a value between 1 second and 10 seconds chosen randomly at every key press. What Applescript could make the trick?  Also, where can I find a list of all codes for each key of the keyboard? What's the "Bad magic number" ImportError in python, and how do I fix it?  The only thing I can find online suggests this is caused by compiling a .py -> .pyc file and then trying to use it with the wrong version of python. In my case, however, the file seems to import fine some times but not others, and I'm not sure why.  The information python's providing in the traceback isn't particularly helpful (which is why I was asking here...), but here it is in case it helps:  Traceback (most recent call last):   File "run.py", line 7, in &lt;module&gt;     from Normalization import Normalizer Resize (enlarge) an image without resampling (in favor of quality loss)? How to resize an image or selection without resampling the pixels? I have some pixel art that I want to enlarge without losing the hard, non-anti-aliased, pixel edges. In other words, 1px should become 4px of the same color— fake big pixels. Is there any way to achieve this procedurally?  Example (of what I don't want):  before resize:    after resize:    Open to other tools/methods beyond photoshop, thanks in advance. Updating global js variable within function and sending updated to HTML document How do I install/update drivers on Debian 9? I just installed Debian 9 on my laptop, however Wifi isn't working and I'm not sure if my graphic card is either. I'm sure that it's just a lack of drivers, but I've never actually had to update drivers on Linux before. So how do I do that? Pointwise convergent and total variation I'm preparing for a test for real analysis and I came across this problem in Royden's book:  Let $\{f_n\}$ be a sequence of real valued functions on $[a,b]$ that converges pointwisely on $[a,b]$ to the real valued function $f$. Show that $TV(f) \leq \liminf  ~TV(f_n)?$  This looks quite similar in form to Fatou's Lemma to me, but can't find any way to establish TV with integration, can anybody please help?  (TV is short for total variation) For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$? Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|&lt;1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that $$\left|\sum_{i=1}^n a_iz_i\right|&lt;\sqrt3.$$ (Disclaimer: I've read the  and I do understand that this is bordering on off-topic since it's about consumer devices. However I hope that I've managed to make enough on-topic by asking about why it is the way it is, and what is the explanation from an electromagnetic point of view - in other words, what laws of electricity and what practical considerations have gone into these decisions).  I recently bought a power tool and in its manual I found a few peculiar safety rules. I cannot see why they would be there, although reading up more it seems that they are commonplace. My understanding of electricity is limited (just about enough to be dangerous), so I'm hoping to expand it. :)  The rules are:   Don't use an extension cord longer than 25m. Considering this is an outdoors tool, this limitation is severely restrictive. The best explanation I've seen is that longer cables would simply have too high of a resistance and the tool wouldn't be able to operate at full power. But this argument doesn't convince me. The tool is rated at 2.4kW, and it's meant to be used with 220V AC power source, which means it uses about 11 amps of current. All the extension cords I've ever come across have at least 16A limitation, so there is quite a reserve still. In addition, the power has to travel much, much further to get to my house, and even after the meter it still takes probably close to those same 25m to get to the power outlet. Why only 25m after the outlet? The resistance of the cables isn't that big, is it? Don't daisy-chain extension cords. I suspect this is a variation of the above - don't exceed 25m; every connection adds additional contact resistance; water in the outdoors connections can be a problem - in general, it just adds risk. However, apart from the water in the contacts, the rest seems to me to be practically negligible. Is there really a significant contact resistance? Assuming that the extension cords are in a good shape. Make sure that the power cord is entirely unrolled. This is another odd one. My suspicion is that it has to do with inductive resistance. But is it really that strong? Wherever I've seen people actually wanting to create an electromagnet, it's pretty much hard work. You need a lot of neat, tightly-wound loops of wire. An extension cord has a lot of insulator around it, spacing the wires pretty far apart, and the loops are very chaotic. In addition, if that was a problem, why do  exist? They're used especially when you need a very long extension cord, and you don't need to unroll them completely every time you want to use them. Solved: OK, I was off the mark. The correct answer (also a ) has to do with the wire heating up from being used close to its limits. When coiled together, the temperature can go pretty high and melt the cables. Not fun. I am looking to build a camera trap for wildlife viewing, able to capture pictures (or even videos) in decent quality to be able to identify distinct features. I do however need to be able to see the pictures in real time, so the camera would need some sort of wireless connection to a receiving device such as a phone.  I thought of using an FPV setup as used in drone racing, but the quality and range seems to be quite low, and the strengths of an analog signal seem to be latency and fps, which are both not important to me. It would be nice to have some sort of digital signal to be able to do some editing, such as storing pictures. It would also be cool to connect several cameras to one device.  Are there any solutions to this problem? Would a raspberry pi/arduino setup work in some way? Thanks! Geodesics equations via variational principle I would like to recover the (timelike) geodesics equations via the variational principle of the following action:  $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$  Using an arbitrary auxiliary parameter $\lambda$, then one is able to rewrite the action and obtains: $$ d\tau = \sqrt{-g_{\mu\nu}\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$ $$ -m \int d\tau = -m \int (d\tau / d\lambda) d\lambda =  -m\int\sqrt{-g_{\mu\nu}\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$  Now we can vary the paths $x^{\mu}(\tau) \rightarrow x^{\mu}(\tau) + \delta x^{\mu}(\tau)$.  I need the following equation to be true, so i get the geodesics equations, but I don't know how one can say the following: $$ \int d\tau \,\delta g_{\mu\nu}\frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = \int d\tau\, g_{\mu\nu, \rho} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} \delta x^{\rho} $$ I know there is a simpler action, which leads to the same equation of motion, but there one has to do the same manipulation, which I sadly don't understand.  Thanks for helping me! Is it "legal" to CREATE and DROP #SomeTable more than once? Git error: large files "detected", even though no large files exist Full URL not working with php include Which is correct: "one or more is" or "one or more are"? Should the phrase be "one or more is...", or "one or more are..."? I'm struggling with some concepts in my algebra class. I've chosen these two completely unrelated examples that demonstrate my struggling topics (characteristics and function rings) the two are the following:  $1$) Let $X$ be a set, and let $S$ be a ring. Give a proof that the left distributive law holds for $(S^X, +, *)$.  $2$) Suppose $Y^3 = Y$ for each element $Y$ of a unital ring $R$. Show that $R$ has a finite characteristic d that is a divisor of $6$.  Any example solutions for these proofs would be helpful. Thank you. Why don't players get extra xp for large encounters? Based on the table Encounter XP Multipliers from the Basic Rules' DM section, Page 57, when you're building encounters in 5E, you include an XP modifier depending on the number of creatures your PCs will be fighting: 1.5x for a pair, 2x for 3-6 and so on. However, the PCs only get the base XP as a reward.     This doesn't change the actual xp reward ... just your calculations of how difficult the encounter is.   Why is this? After acknowledging that battles vs multiple opponents are more difficult, why not reward them as such? What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? Are GUID collisions possible? I'm working on a database in SQL Server 2000 that uses a GUID for each user that uses the app it's tied to. Somehow, two users ended up with the same GUID. I know that microsoft uses an algorithm to generate a random GUID that has an extremely low chance of causing collisons, but is a collision still possible?