Proving that cartesian product is a group
Proving a direct product of groups is a group
Is stats maths or not?   Given that it's all numbers, mostly taught by maths departments and you get maths credits for it, I wonder whether people just mean it half-jokingly when they say it, like saying it's a minor part of maths, or just applied maths.   I wonder if something like statistics, where you can't build everything on basic axioms can be considered maths. For example, the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles. Subscribing to questions and comments that don't belong to you How to link a variable of a material to another material? What are the specifics of 18.04's ten year support? Cu has an unpaired electron in 4s, but it is diamagnetic. I thought that it has to be paramagnetic. What am I missing? Simple regular expression for a decimal with a precision of 2 Is the following proof correct, or are certain steps illegal?  Suppose $f:G_1\to G_2$ is a surjective group isomorphism, $N_1\lhd G_1, N_2\lhd G_2$ normal subgroups, with $N_1=f^{-1}(N_2)$. Prove that $(G_1/N_1)\cong (G_2/N_2)$.  Proof:  As $f$ is surjective, we an use the first isomorphism theorem to get $G_1/\ker(f)\cong G_2$, which implies that $$(G_1/\ker f)/(N_1/\ker f)\cong G_2/(N_1/\ker f)$$  Using the first isomorphism theorem we also know that $$N_1/\ker (f)\cong f(N_1)=f(f^{-1}(N_2))=N_2$$  Combining this we can see that $(G_1/\ker f)/(N_1/\ker f)\cong G_2/(N_1/\ker f)\cong G_2/N_2$.  Using the third isomorphism theorem we have that: $$ (G_1/\ker f)/(N_1/\ker f)\cong G_1/N_1$$  And from the last two lines we have $G_1/N_1\cong G_2/N_2$ $\tag*{$\Box$}$ I'm printing a tradeshow flyer and was wondering what type of paper I should print it on. I don't want anything glossy so I'm trying to decide on the following:   100# text weight matte 80# cover 80# matte   What is the difference between the types above? And how can I make a decision? Does adverb placement affect meaning? He swam slowly to the island.  He slowly swam to the island.    Some experts say that there is a “slight difference” in meaning. Would you please tell me that difference? A inequality proposed at Zhautykov Olympiad 2008 An inequality proposed at Zhautykov Olympiad 2008.     Let be $a,b,c &gt;0$ with $abc=1$. Prove that:    $$\sum_{\mathrm{cyc}}{\frac{1}{(a+b)b}} \geq \frac{3}{2}.$$   Set $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x}$.   Our inequality becomes: $$\sum_{\mathrm{cyc}}{\frac{z^2}{zx+y^2}} \geq \frac{3}{2}.$$  Now we use that: $z^2+x^2 \geq 2zx.$ $$\sum_{\mathrm{cyc}}{\frac{z^2}{zx+y^2}} \geq \sum_{\mathrm{cyc}}{\frac{2z^2}{z^2+x^2+2y^2}} \geq \frac{3}{2}.$$  Now applying Cauchy-Schwarz we obtain the desired result.  What I wrote can be found on this link: . But now, I don't know how to apply Cauchy-Schwarz.   Thanks:) Exclude one pattern from glob match Do spice containers need to be hermetic? Spice containers purchase  I'm planning on upgrading my spice storage. Reading some advice online (e.g. , and ) I came to the conclusion that spice containers should:   Not be exposed to direct sunlight and high temperatures Be airtight   Which means that either you store it inside of some closed cupboard/wardrobe/room or you have containers which do block light and are in a relatively shady place. And the containers need to be airtight.  Available solutions  I've been browsing for some containers available online in my country and the 2 most sensible options I've seen are these:       The problem  The first one is hermetic (it has some rubber around the lid) but it's clear glass. The other one does not have the rubber, just a regular twist-off lid, but it's amber glass.  It'll be hard for me to find any space inside of cupboards to store all of my spices (I need at least 30), so I'd rather buy amber glass container, which I believe will effectively block the sunlight, even when kept on the counter. But I can't find any hermetic amber-glass containers, only ones with regular lid.  The question  Is that regular twist-off lid enough to keep spices airtight and fresh? Or should I focus on buying a hermetic container (like the one in clear glass). I could then for example paint them black to block the sunlight. When programming interfaces, I've found I'm doing a lot of casting or object type conversion.  Is there a difference between these two methods of conversion?  If so, is there a cost difference or how does this affect my program?  public interface IMyInterface {     void AMethod(); }  public class MyClass : IMyInterface {     public void AMethod()     {        //Do work     }      // Other helper methods.... }  public class Implementation {     IMyInterface _MyObj;     MyClass _myCls1;     MyClass _myCls2;      public Implementation()     {         _MyObj = new MyClass();          // What is the difference here:         _myCls1 = (MyClass)_MyObj;         _myCls2 = (_MyObj as MyClass);     } }   Also, what is "in general" the preferred method? How do I remove Steam? Anyone know of a way to remove Steam from Ubuntu 12.04? Looked around the net but could not find an straight forward solution.    Thanks. How do I get Siri to pronounce my name correctly? Siri doesn't pronounce my name correctly. I tried editing the phonetic spelling as it says on many sites, but it doesn't fix the problem. How do I fix this problem? Is it possible to build a .dmg file (for distributing apps) from a non-Mac platform? And if yes, how? I have recently found an argument against disabling a root user login in Linux at   I assume that, if everybody uses a public key authentication, there's no risk at losing the root password.  Is it always better to disable the root login via ssh? What are the benefits to marking a field as `readonly` in C#? What are the benefits of having a member variable declared as read only?  Is it just protecting against someone changing its value during the lifecycle of the class or does using this keyword result in any speed or efficiency improvements? What is the progression route in Terraria? It appears that Unity on Raring only comes with a single virtual desktop (aka workspaces) by default, and CCSM's Wall plugin has no way to add more. Is this configurable at all?