Inclusive mixin with JS
How can I merge properties of two JavaScript objects dynamically?
Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers & UFD I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1):  Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta=\varepsilon\gamma^n$, for $\alpha,\beta$ relatively prime numbers and $\varepsilon$ a unit, implies $\alpha=\varepsilon^{\prime}\xi^n$ and $\beta=\varepsilon^{\prime\prime}\eta^n$, with $\varepsilon^{\prime},\varepsilon^{\prime\prime}$ units.  My attempt: once $\alpha$ and $\beta$ are relatively prime, $\alpha\nmid\beta$ and $\beta\nmid\alpha$. Therefore $N(\alpha)$ and $N(\beta)$ are relatively prime too (where $N(a+ib)=a^2+b^2$). Therefore,  $$N(\alpha\beta)=N(\alpha)N(\beta)=N(\varepsilon)N(\gamma^n)=N(\gamma)^n=(p_1\cdot\ldots\cdot p_r)^n=p_1^n\cdot\ldots\cdot p_r^n$$  And once $N(\alpha)$ and $N(\beta)$ are relatively prime, we have (if necessary, we reorder the $p_i$'s):  $$N(\alpha)=p_1^n\cdot\ldots\cdot p_k^n=(p_1\cdot\ldots\cdot p_k)^n=N(\xi)^n=N(\xi^n)$$ and $$N(\beta)=p_{k+1}^n\cdot\ldots\cdot p_r^n=(p_{k+1}\cdot\ldots\cdot p_r)^n=N(\eta)^n=N(\eta^n)$$  And the result follows.  But I'm not sure about one step. It is $\alpha$ and $\beta$ relatively prime $\Rightarrow$ $N(\alpha)$ and $N(\beta)$ relatively prime. I know that if $\alpha|\beta$ then $N(\alpha)|N(\beta)$, but the previous assertion is not necesseraly true, and if it is, and don't know how to prove it. If it is true, the result follows, unless I made something wrong in the rest of the proof.  That's it. If you know another way of doing the exercise, please show me. "a ten years old boy" or "a ten year old boy" Where should I put software I compile myself? Kali Linux: apt update returns "Hash Sum mismatch" error I'm experiencing exactly the same issue as described in this question: . Before you mark this as a duplicate however, I have tried the solutions posted there, as well as on numerous other sites, including:  sudo apt-get clean sudo rm -rf /var/lib/apt/lists/* sudo apt-get update   Editing /etc/apt/sources.list with alternate official mirrors, such as  deb http://mirrors.ocf.berkeley.edu/kali kali-rolling main non-free contrib or deb https://http.kali.org/kali kali-rolling main non-free contrib  Everything worked after I first imported the VM. I ran sudo apt update and it found some ~650 packages to upgrade. I ran sudo apt upgrade and it encountered an error partway through. That error was solved using sudo apt --fix-broken install, but that is when this hash sum error began. Unfortunately due to hours of troubleshooting I no longer have the details of the earlier error, but I believe it was an error extracting a package due to corrupt data.  I've tried multiple official mirrors, but I get the same error. Additionally, when I downloaded the Packages.gz file  on my Windows machine (VM host) and computed the SHA256 hash, I got the exact hash that apt printed as the expected value. This led me to believe that the error was not with the mirror but with my VM.  The next thing I tried was wget https://mirrors.ocf.berkeley.edu/kali/dists/kali-rolling/main/binary-amd64/Packages.gz followed by sha256sum Packages.gz, which provided yet another different hash output.  To be clear, I have seen 3 different hashes for the same file:   The "correct" hash shown by apt as expected, which is the one that windows also produced after downloading the file using a browser The incorrect hash calculated by apt, which led to the error A different hash calculated by sha256sum after downloading the file using wget using the same URL as for the browser download   I should also note that I have only been referencing the SHA256 hash in each step. The other hash functions are also mismatched when I run sudo apt update, but the file size is the same. I had considered that downloads might be failing due to limited disk space (it is a VM after all) but I don't think that's the case.  What am I missing? Can I just change straightforward wheel axles to QR skewers? A machine is described as being "really slow" or "freezing". How can I troubleshoot/resolve this issue? What should be the first steps to take? Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even I have a question:     Suppose $f$ is continuous and even on $[-a,a]$, $a&gt;0$ then prove that   $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$   How can I do this? Don't know how to start. Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you. When is it OK to shrink a Database? I know shrink is the devil: It reverses page order and is responsible for skin cancer, data fragmentation, and global warming.  The list goes on... That being said, say I have a 100 GB database and I delete 50 GB of data -- not on one table, but a general pruning of old data on a database wide level, covering 90% of the tables -- does this constitute an appropriate use case for shrinking the database?  If not, what are the appropriate steps to take to clean house after removing such a high percentage of data from a database?  I can think of two: Rebuild Indexes and Update Stats. What else? How can I pass a Parameter via Query String to populate a lookup field on a New Item Form/DVWP? Suppose a particle only influenced by gravity is going from point a to point b on an x,y plane. Is there an equation in which the particle takes the longest amount of time, yet still makes it to point b? I know the equation for brachistochrone is the parametric for an inverted cycloid, yet I'm not sure of the opposite. For reference, I am a high school student wanting to try this for a research project. I really only need to know whether this has already been done or not, but attempted solutions are obviously appreciated. Is there a limit as to how fast a black hole can grow? This is a homework question of mine that Ive answered and I'd really like some feedback on my solution.       Prove or find a counter example to the claim that for all sets $A,B,C$, $$\text{if  }A\cap B=B\cap C=A\cap C=\emptyset \text{ then } A\cap B\cap C\neq \emptyset$$.   Since I can only either prove it is true, OR find a counter example I have gone with the latter because I think this statement is not true. My counter example is as follows:  Let $A = \{1,2,3 \}$ , $B = \{4,5,6 \}$ and $C = \{7,8,9 \}$. It is then clear that $A ∩ B =  B ∩ C =  A ∩ C = \emptyset$ .  If we let $x ∈ A\cap B\cap C$ then $x$ must be in $A$ and $B$ and $C$. Since this cannot be possible because there are no common elements then this implies that $A∩B∩C$ has to be equal to a null set i.e $A∩B∩C = \emptyset$ and thereby showing that (at least for this scenario) if $A \cap  B =  B \cap  C =  A \cap  C = \emptyset$ then $A\cap B\cap C = \emptyset$.  Is this okay? Any feedback will be greatly appreciated thank you! Is there an official Marvel Cinematic Universe viewing order? I'm planning on re-watching the currently-released 5 Marvel films prior to seeing The Avengers.  Have Marvel released any sort of official viewing order?  Captain America: The First Avenger seems like the hardest one for me to place in order with the rest.  It takes place prior to the other films, but has the greatest number of references to the other films.  So should it be watched first, or last?  The rest of them are relatively clear, as they're chronological in the order they're released, with the possible exception of The Incredible Hulk and Iron Man 2 happening at the same time. Why are orbits elliptical? Texmaker GUI is messy with high resolution screen Move object in inches instead of feet? Whenever I move an object in blender in unit mode(Imperial), it always make it in feet. For example, if I type G Y 12, it will move 12 feet. How can I make it move 12 inches instead? 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