Choosing lens for a studio to record classes
What is the formula for percent of frame filled at a specific distance and focal length?
How do you implement Coroutines in C++ Bypass GPG signature checks only for a single repository What is the metal sheath surrounding a USB connector, and is it commonly connected? I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. Can you partition an infinite set, into an infinite number of infinite sets? I've installed Ubuntu 16.04 on my Asus X555LJ.  But I don't know how to install drivers or how to enable function keys to controls screen backlight brightness. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ I haven't been able to do this exercise:     Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse   image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.   Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.  Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.  Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.  If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.    What do you think? What are nodes, and how do I use them? I'm a bit new to Blender, and in trying to learn more, I've stumbled across nodes.  What are nodes, and how do I use them? 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! Mean or sum of gradients for weight updates in SGD Encode/Decode URLs in C++ Does anyone know of any good C++ code that does this? Can I use a higher rank tensor field as the metric field? What a metric should necessarily do is it should give me a way to associate a frame invariant number with a given pair of spacetime events. Now, if I use a higher rank tensor field (say, for example, a tensor field of rank 3: $g_{\mu \nu \rho}$) then also I can certainly produce a frame invariant scalar out of a given displacement $\vec{A}$ in this trivial way: $ I := g_{\mu \nu \rho} A^{\mu}A^{\nu}A^{\rho}$. Since the components of the higher rank tensor and that of the displacement vector are going to transform in a covariant and a contravariant manner respectively, the cooked up quantity is certainly a scalar.   Another crucial property, which I think has more direct physical content than the previous one, is that between two inertial frames, at least one such transformation should exist that leaves at least one metric invariant. i.e., There should exist at least one combination of transformation matrix $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}$ and metric $g_{\alpha \beta \gamma}$ that satisfies the following equation:      $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}\displaystyle\frac{\partial{x^{\beta}}}{\partial{x^{\nu '}}}\displaystyle\frac{\partial{x^{\gamma}}}{\partial{x^{\rho '}}} g_{\alpha \beta \gamma} - \delta_{\mu '}^{\alpha}\delta_{\nu '}^{\beta}\delta_{\rho '}^{\gamma} g_{\alpha \beta \gamma} =0$    The last thing I can think of that can put a restriction on the choice of a tensor as a metric is the existence of a possibility of finding a metric compatible symmetric connection field. Following the usual procedure of finding the expression for a metric compatible symmetric connection field, I reached following condition (unlike the case of the usual two rank metric where we get a full-fledged expression) for the connection in the terms of the metric:      $g_{\nu \rho k_1} \Gamma^{k_1}_{\mu \lambda} - g_{\lambda \mu k_2} \Gamma^{k_2}_{\rho \nu} = \displaystyle\frac{1}{2} (\partial_{\nu}g_{\rho \lambda \mu} + \partial_{\rho}g_{\lambda \mu \nu}  - \partial_{\lambda}g_{\mu \nu \rho}  - \partial_{\mu}g_{\nu \rho \lambda}) $   My question is that if it is possible to satisfy the two highlighted conditions then can we use such a 3 rank (or even higher rank tensors with similarly produced conditions) tensor fields as metric fields?   PS: This is NOT a proposal for a new home-production theory of gravity (or that of anything for that matter) but rather it is just that I am trying to understand why a two rank tensor is used in General Relativity as the metric. Thank you. "Do the needful" -- Why is it used instead of asking a question? The plugin generated X characters of unexpected output during activation (WordPress) What do damage buffs affect? In Overwatch, there are several ways to buff the damage dealt by a hero: the alternate fire of Mercy's Caduceus Staff, Ana's Nano Boost, and Orisa's Supercharger. Are they all governed by the same rules? What can they affect, or not affect?  This question exists to have a single reference point for the many, many questions that have been asked, instead of having to update each answer individually when game mechanics change Erase Windows Vista to Install Ubuntu new here. I have an old E-machine which as Windows Vista on it and I want to erase the Windows Vista, and install Ubuntu on it instead, but I am not sure where to start. Can I first install Ubuntu, and THEN erase Windows Vista, or is there a more sure method? I appreciate your help. After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again. Is that normal?  Also, can I set a newly added user as a sudoer or do I have to change that only after adding it? How can I do that via the shell?  Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? How to share files through the local network? I have a desktop (Ubuntu 13.04 &amp; Win 7 Home Premium) and a Laptop (Ubuntu 12.04), both having WiFi adapters. I also have a WiFi router connected to internet which both my computers can access.  I want to share files between my desktop and my laptop using WiFi (similar to Homegroup on Windows 7 but without using Ethernet cables).  How do I set-up that?  I want to share files using both OSs, if anyone have have information about sharing files with any of the OS please answer! I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? NSString tokenize in Objective-C