How to compute $\prod_{1\le i\le n} \left(1-\frac{1}{2^i}\right)$
Limit of a particular variety of infinite product/series
MacBook Pro 2020 Fan Issues I recently purchased a MacBook Pro 2020 13&quot; (1.4 GHz, 8GB Ram, i5). For some reason, when I open Chrome and watch Twitch my computers fans turn on and go crazy. After opening Activity Monitor, I noticed that Google Chrome Helper (Renderer) uses around 50-80% of my CPU. Also, sometimes when I run code on my MacBook Pro, the fans turn on when the program is compiling/running but slow down a few minutes after terminating the program. Also, when watching Twice on Safari, the fans don't come on. However, around 50-60% of the CPU is still being used. Should I be concerned about the CPU usage (I am nervous because I only recently purchased the laptop)? Turn off close Facebook friend notification Looking for a Pre-1975, young adult, sci-fi, colored circle dimension crossing Looking for the title and or author of a long ago favorite; I have searched the web for it on and off for years.  I read a paperback novel about 40 years ago; it was probably a used book at that time. Probably pre-1975 publication.  In my memory it was written in the style of Andre Norton I have looked through her , but I am not finding it. It was a young adult sci-fi novel.  Plot; a young woman (gender was not strong, but I remember the main character being female) can see different colored circles floating around. When she looks at them just right, she can jump through to another universe.  In the other universe, things change.  She might become a lion, or maybe she is carrying a seed that becomes a net in the other universe.   Two bad guys are chasing her around the different dimensions, I recall one scene where they are all sliding down an icy slope and she jumps through a portal, to escape the bad guys.   There were a lot of complexities about, dimensions, what became what, different colors of portals and such. How to force a table into page width? I have the following table:  \begin{table}[htb] \begin{tikzpicture} \node (table) [inner sep=0pt] { \begin{tabular}{ l | l }   {\bf Symptom} &amp; {\bf Metric} \\ \hline Class that has many accessor methods and accesses a lot of external data &amp; ATFD is more than a few\\ Class that is large and complex &amp; WMC is high\\ Class that has a lot of methods that only operate on a proper subset of the instance variable set &amp; TCC is low\\ \end{tabular} }; \draw [rounded corners=.5em] (table.north west) rectangle (table.south east); \end{tikzpicture} \caption{God class symptoms} \label{tbl:god_class} \end{table}   Now I want to force the width of the table to be the same as the \textwidth, either by linewrapping of table text or by scaling. How can I achieve that? Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$    I think this is known (see ), I appreciate any hint or link for the solution (or the full solution). What is the relationship between symmetry and degeneracy in quantum mechanics? Let me remind you about the following classical  examples in quantum mechanics.     Example 1. Bound states in 1-dim potential V(x).   Let $V(x)$ be a symmetric potential i.e.   $$V(x) = V(-x)$$   Let us introduce the parity operator $\hat\Pi$ in the following way:   $$\hat\Pi f(x) = f(-x).$$   It is obvious that    $$[\hat H,\hat\Pi] = 0.$$   Therefore, for any eigenfunction of $\hat H$ we have:   $$\hat H|\psi_E(x)\rangle = E|\psi_E(x)\rangle = E\hat\Pi|\psi_E(x)\rangle,$$   i.e. state $\hat\Pi|\psi_E(x)\rangle$ is eigenfunction with the same eigenvalue. Is $E$ a degenerate level? No, because of linear dependence of $|\psi\rangle$ and $\hat\Pi|\psi\rangle.$   Consider the second example.     Example 2. Bound states in 3-dim a potential $V(r)$. Where $V(r)$ possesses central symmetry, i.e. depends only on distance to center.   In that potential we can choose eigenfunction of     angular momentum $\hat L^2$ for basis    $$|l,m\rangle,$$   where $l$ is total angular momentum and $m$ - its projection on chosen axis (usually $z$). Because of isotropy eigenfunction with different $m$ but the same $l$ correspond to one energy level and linearly independent. Therefore, $E_l$ is a degenerate level.      My question is if there is some connection between symmetries and degeneracy of energy levels. Two cases are possible at the first sight:   Existence of symmetry $\Rightarrow$ Existence of degeneracy Existence of degeneracy $\Rightarrow$ Existence of symmetry     It seems like the first case is not always fulfilled as shown in the first example. I think case 1 may be fulfilled if there is continuous symmetry. I think the second case is always true. Add and remove update channels in an easy terminal way I know there is a GUI in Ubuntu Software &amp; Updates to enable the update channels   updates proposed backports security   as shown in this screenshot:    I am looking for an easy way to do this from within a terminal using commands such as    sudo apt-add-update enable updates sudo apt-add-update enable proposed sudo apt-add-update enable backports sudo apt-add-update enable security  sudo apt-add-update disable updates sudo apt-add-update disable proposed sudo apt-add-update disable backports sudo apt-add-update disable security   and an additional thing  sudo apt-add-update enable default  sudo apt-add-update disable default     Some examples for a better understanding   An empty sources.list   cat /etc/apt/sources.list  &lt;empty&gt;  sudo apt-add-update enable security  &lt;empty&gt;   One enabled repository (main)   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily-security main   Two or more enabled repositories in one or two lines   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe   or   deb http://archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily universe  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu wily-security main universe   or  deb http://archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily-security main deb http://archive.ubuntu.com/ubuntu wily universe deb http://archive.ubuntu.com/ubuntu wily-security universe   With deb-src entries   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe deb-src http://archive.ubuntu.com/ubuntu wily main universe  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe deb-src http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu wily-security main universe deb-src http://archive.ubuntu.com/ubuntu wily-security main universe   With inactive deb-src entries   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe # deb-src http://archive.ubuntu.com/ubuntu wily main universe  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe # deb-src http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu wily-security main universe   The default thing    cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily-security universe  sudo apt-add-update enable default  deb http://archive.ubuntu.com/ubuntu wily universe deb http://archive.ubuntu.com/ubuntu wily-security universe   Only one entry and the disable action   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily-security universe  sudo apt-add-update disable security  &lt;empty&gt;   Different or the same servers for different or the samerepositories, respect each server   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily universe deb http://us.archive.ubuntu.com/ubuntu wily main  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily universe deb http://us.archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily-security universe deb http://us.archive.ubuntu.com/ubuntu wily-security main   Different Ubuntu releases for different repositories, respect each release   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu trusty main  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu trusty main deb http://archive.ubuntu.com/ubuntu wily-security main universe deb http://archive.ubuntu.com/ubuntu trusty-security main   PPAs or other package sources (not-Canonical)  in the sources.list?  Ignore! Don't change the protocols, e.g. https, http, tor, ... How many squares on a 5-foot grid does a 5-foot radius spell affect? We had a situation come up regarding this term and how it affected spells.  In the spell Moonbeam, for example, it says a 5-foot radius cylinder appears. Does this mean that from the center of square it extends 2.5 feet into the next square hypothetically hitting up three squares or does this mean from the edge of one the target square to the five feet and five feet the other direction effectively covering only two squares for a total of 10 feet between total range of both squares?  I understand that radius means half the distance of a circle from the center point. The spell is not the issue but the understanding of radius as it pertains to the spell, as there are other spells that are worded similarly such as Fireball, etc.  Essentially we are asking, does the spell in the example effect one square or three? This would be saying the edge of the initial square is the point of origin and not the center of the square for the total distance of ten feet.  I had been working under the understanding that the radius is half of the distance and therefore dragging a ruler from the center of where I want the spell to take effect would get me the area of effect.  It's much easier to do this with Fireball, due to the larger area to effect, but this one has such a small area and therefore sparked the debate. We are using a grid system to mark the distance, etc. We play within the digital realm of roll20.com. I recently purchased a MacBook Pro 13&quot; with the M1 chip and transferred all my data over using a Time Machine backup. Homebrew packages are currently installed in /usr/local/opt/ but it is my understanding that they should be under /opt/homebrew for Apple Silicon Macs. How can I:  Get the list of packages I currently have installed Remove those from /user/local/opt Reinstall them under /opt/homebrew  I kind of assumed Homebrew would be clever enough to do this automatically but apparently it needs some manual maintenance. Here's my Homebrew config if it helps: HOMEBREW_VERSION: 2.7.2 ORIGIN: https://github.com/Homebrew/brew HEAD: dad7dc6a1498b80770d98f2d7cd6fb927c300bbb Last commit: 3 days ago Core tap ORIGIN: https://github.com/Homebrew/homebrew-core Core tap HEAD: 2b1d79d038bffdbfcee93283051d48919a4caf3d Core tap last commit: 16 hours ago Core tap branch: master HOMEBREW_PREFIX: /usr/local HOMEBREW_REPOSITORY: /usr/local/Homebrew HOMEBREW_CELLAR: /usr/local/Cellar HOMEBREW_CASK_OPTS: [] HOMEBREW_MAKE_JOBS: 8 Homebrew Ruby: 2.6.3 =&gt; /System/Library/Frameworks/Ruby.framework/Versions/2.6/usr/bin/ruby CPU: octa-core 64-bit arm_firestorm_icestorm Clang: 12.0 build 1200 Git: 2.30.0 =&gt; /usr/local/bin/git Curl: 7.64.1 =&gt; /usr/bin/curl macOS: 11.1-arm64 CLT: 12.3.0.0.1.1607026830 Xcode: N/A Rosetta 2: false  Thanks in advance! Is Duck/Duct tape safe to use to insulate mains wires? Question says it all really. Based in the UK so mains is 240v and 20 amps. It's a lighting circuit. Find all solutions to $x^9 \equiv 25 \pmod {29}$ How to draw a picture by using tikz as backgroud of a character? I am writing some Chinese character and hope to draw simple pictures by tikz and use it as background of this chinese characters. In a short I'd like to get some results as this picture (ignore the lower right noise):    How could I do? Thanks in advance. As an example for least workable code:  \documentclass{article} \usepackage[CJKnumber]{xeCJK} \setmainfont{KaiTi}  \begin{document} \Huge 明月出天山 \end{document} Xcode 6.1 won't let me create a new project I'm running Xcode 6.1 on OS X Yosemite and I'm unable to create a new project, as you can see below it just says 'loading'.  I've removed the product and org name for the purpose of this image. I have tried uninstalling but that didn't work. I am new to Xcode and coding in general so I would appreciate it if any solution could be given in lay terms. How can I rotate an object based on another's offset to it? I have a 3D model of a turret that con rotate around the Y-axis. This turret has a cannon that is significantly off the center of the object. I want the cannon, not the turret, to aim at a specified target. I can only rotate the turret, however, and thus I don't know what equation I need to apply in order to accomplish by objective.  The following image illustrates my problem:  If I have the turret "LookAt()" the target, a laser originating from the cannon will completely miss said target.   If this were a completely top-down scenario, and the cannon were exactly parallel to the turret, then my logic tells me that the fake target should be located at a position that's equal to the actual target plus an offset that's equal to that between the turret and the cannon. However, in my actual scenario, my camera is angled 60º, and the cannon has a slight rotation.  The following image illustrates the scenario:   I'm not sure exactly why, but if I apply that same offset, it only seems to work while aiming at certain distances from the turret.  Is my logic flawed? Am I missing something fundamental here?  Final Edit: the solution provided by @JohnHamilton latest update solves this problem with perfect precision. I have now removed the code and images that I used to illustrate my incorrect implementations. What's the -alhF flag in ls? What is the -alhF flag in ls? I can't find it in the . How do I convert a SMP server world into a Single-player compatable world? I had a Minecraft world on Minefold.  I archived my Minecraft world just before Minefold closed about a year ago.  I'm trying now to open the world in my local Minecraft installation on OS X, but it doesn't show up in the worlds list.  To import it, I simply unzipped the archive from Minefold into my ~/Library/Application Support/minecraft/saves folder.  I'm running Minecraft 1.8, and I don't have any mods installed.  There's no error; the world just doesn't show up in my list of worlds after I click [Singleplayer] (or [Multiplayer]).  Now, the big clue is that the directory structures are different between my locally-created worlds and the world created on Minefold, but I can't figure out how to adapt the files to make them match:  This is a screenshot of my saves folder. I've circled in blue "local" and "minefold", which contain data files for the local and Minefold worlds, respectively.    I bracketed similar filenames in both worlds, but I'm flummoxed in that the Minefold version has three (3) sets of data files, whereas the local world has only one (1) set of files (even though I have visited Nether in the local world).  How do I load my old Minefold world locally? If $$S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}\,$$ find the value of $14S$.  The question can be simplified to:  Find $S=\sum\limits_{k=1}^n\,t_k$ if $t_n=\dfrac{n}{1+n^2+n^4}$.  As $1+n^2+n^4$ forms a GP, $$t_n=\frac{n(n^2-1)}{n^6-1}\,.$$ But I can't figure out how to solve further. It would be great if someone could help. 3D graphics render as though no illumination Can I gift a game via the PlayStation Network? Is it possible to "gift" a game (e.g., Castle Crashers) to someone through the PlayStation Network?  I don't have a PS3 but was considering buying Castle Crashers as a gift for a friend so that we could play co-op.  Is this possible? find exec '{}' not available after > Exec allows us to either pass all arguments at once with {} + or to pass them one by one with {} \;  Now let's say I want to rename all jpeg, no problem doing this:  find . \( -name '*.jpg' -o -name '*.jpeg' \) -exec mv '{}' '{}'.new \;   But if I need to redirect output, '{}' isn't accessible after redirection.  find . \( -name '*.jpg' -o -name '*.jpeg' \) -exec cjpeg -quality 80 '{}' &gt; optimized_'{}' \;   This wouldn't work. I'd have to use a for loop, storing find's output into a variable before using it. Let's admit it, it's cumbersome.  for f in `find . \( -name '*.jpg' -o -name '*.jpeg' \)`; do cjpeg -quality 80 $f &gt; optimized_$f; done;   So is there a better way?