A universe from nothing
Is the total energy of the universe zero?
Given $a_1, a_2, a_3, \ldots , a_n$ not all zero, show $\gcd(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$. Also deduce $a_1x_1+a_2x_2+ \ldots +a_nx_n = b$ has solutions $\iff \gcd(a_1, a_2, a_3, \ldots , a_n)|b$.   I know that $ax+by=n$ has integer solutions $\iff \gcd(a,b)|n$. I'll need to use induction to show $a_1x_1+a_2x_2+ \ldots +a_nx_n = \gcd(a_1, a_2, a_3, \ldots , a_n)$ has solutions, and then I will be able to say that as $\gcd(a_1, a_2, a_3, \ldots , a_n) | a_1x_1+a_2x_2+ \ldots +a_nx_n $ for any $x_1,x_2, x_3 \ldots x_n$ then $c \leq \gcd(a_1, a_2, a_3, \ldots , a_n)$ and $c \geq \gcd(a_1, a_2, a_3, \ldots , a_n)$, and thus $c=\gcd(a_1, a_2, a_3, \ldots , a_n)$. The prime example is C# and .Net. 95% of C# questions are also tagged .Net. The reverse isn't the case so it's not a synonym. Basically, C# is a child of .Net.  Would it be possible to create some sort of one-way linkage like this? The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is , when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d&gt;1$?      — For instance if $p$ is prime (which is not coprime with $0$) then $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &amp;\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} \times \mathbb{F}_{p} &amp;\text{if } p \equiv 1 \pmod 4 \end{cases}$$ (because $-1$ is a square mod $p$ iff $(-1)^{(p-1)/2}=1$).  — More generally, if $n=p_1^{r_1} \cdots p_m^{r_m} \in \Bbb N$, then  each pair of integers $p_j^{r_j}$ are coprime, so that by CRT we get $$\mathbb{Z}[i]/(n) \cong \mathbb{Z}[i]/(p_1^{r_1}) \times \cdots \times \mathbb{Z}[i]/(p_m^{r_m})$$  I was not sure how to find the structure of $\mathbb{Z}[i]/(p^{r}) \cong (\Bbb Z/p^r \Bbb Z)[X] \,/\, (X^2+1)$ when $p$ is prime and $r&gt;1$.  — Even more generally, in order to determine the structure of $\mathbb{Z}[i]/(a+bi)$ with $a+bi=d(x+iy)$ and $(x,y)=1$, we could try to use the CRT, provided that $d$ is coprime with $x+iy$ in $\Bbb Z[i]$. But this is not always true: for $d=13$ and $x+iy=2+3i$, we can't find Gauss integers $u$ and $v$ such that $du + (x+iy)v=1$, because this would mean that $(2+3i)[(2-3i)u+v]=1$, i.e. $2+3i$ is a unit in $\Bbb Z[i]$ which is not because its norm is $13 \neq ±1$.  — I was not able to go further. I recall that my general question is to known what $\mathbb{Z}[i]/(a+bi)$ is isomorphic to, when $a$ and $b$ are integers which are not coprime (for instance $a=p^r,b=0$ or $d=(a,b) = a^2+b^2&gt;1$).  Thank you for your help! How to create a YouTube playlist from a list of links? I have very long lists of YouTube links/bookmarks. I want to create playlists for my YouTube channel with them. Is there any way to do it automatically without having to add each individual URL to a playlist? Consider the sentence &quot;I can run faster than 15 miles per hour.&quot; Its meaning is clear and to my eyes obviously grammatically correct. Now let me present some variations that have given me trouble for a long time.  I am faster than 15 miles per hour. – To me this is clearly incorrect. Directly comparing me to a speed doesn't seem right. We need to compare my speed to a speed, or me to another person.  I can run faster than him. – Compared to the base sentence, there is a distinct shift in meaning of the comparison. While before I named a speed faster than which I can run, now I am naming a person. It doesn't seem quite right. I realize the parts of speech can change, but my initial objection is that &quot;him&quot; is not a speed.  I can run faster than he. – This seems most correct to me, but still somehow feels objectionable. Is this in fact the correct way to say it? And if so, is it proper as is or need I say &quot;... faster than he can&quot; or even &quot;... faster than he can run?&quot;  I am faster than him. – With &quot;am&quot; instead of &quot;can run&quot; it now seems slightly more correct. But is it?  I am faster than he. – I'm in doubt here. It doesn't seem wrong to me to say, &quot;I am faster than he is&quot; or even &quot;I am faster than he is fast.&quot; (Though I suppose that is a given since I could hardly properly compare to some other category as in &quot;I am faster than he is smart.&quot;)  My speed is faster than his. – Hmm. This seems more proper as &quot;my speed is greater than his.&quot;   So which of these constructions is correct and which is incorrect? Is there a general rule that I can follow? UPDATE The scholarly article  may be of great interest to some readers!  Once we accept that the elided constituent and its antecedent can differ in form, it becomes reasonable to ask how large this difference can be. The answer in Rooth (1992), Fiengo and May (1994), Chung et al. (1995) and subsequent work is that the wiggle room is actually quite small: the elided constituent and its antecedent are allowed to differ only in the realization of inﬂectional morphology. Other than that, both constituents have to be syntactically and lexically isomorphic. The new software center in Ubuntu 16.04 shows no application data found This is the error I get when I open the software center. I've tried rebooting and running sudo apt-get update but I see no improvement. It worked earlier but I don't know what went wrong since then. I'm using Ubuntu 16.04 final release on a Dell Inspiron 15R. Any advice will be appreciated. Thanks in advance.  EDIT: The problem magically solved itself without me doing anything special (or maybe inadvertently doing something, I don't know). So I'd still like to get to the root of this. On main and in the old mobile site (iirc) the username of the OP is highlighted in the comments.  This is no longer the case after the new design/theme was rolled out yesterday.  Can we get this back? How do I convert the coordinates of a click to a tilemap position? How do you turn a tilemap (produced by TileD) for cocos2d where I can click on individual items and get a callback?  Is it just a case of converting captured touch and finding out where it is on the tilemap and then converting that to pixels  -- but that doesn't say what item was clicked on.  I have a tilemap of a city with buildings on it, and I'd like to be able to click or press on a specific building, get a callback.  What would be the best way to accomplish this? I saw many saying that the latest version of Ubuntu 16.04 includes an opensource driver for ATI. I have an HP laptop with AMD 6770m + Intel HD 3000 hybrid graphics and wants to install Kubuntu on it. I created a bootable USB drive of Kubuntu 16.04 and booted from it just to check everything works perfectly (I don't want to install it separately and ruin bootloader as of now). But my laptop is getting too hot and the battery is not lasting for more than 30 mins. Also the brightness control is not working at all and it stays high always.   How can I get hybrid graphics on Kubuntu/Ubuntu working well? I am ready to turn off the ATI card since I am not going to play any games on Linux. I just need the Intel graphics + Cool Laptop + Good battery Life + Brightness control.  My BIOS setting for switchable graphics mode is FIXED. Do I need to change it to DYNAMIC?  UPDATE  I wasn't able to make things work in 16.04 so installed 14.04 LTS along with the AMD Catalyst driver. Now everything works perfectly fine. Below are the steps.   Clean install Ubuntu 14.04 LTS Install the fglrx catalyst drivers using “Additional Drivers” window. This will enable us to switch graphics Restart Ubuntu FAN Noise Fix - Refer this  and install i8kutils Brightness Control - Check this     is the official documentation for AMD/Intel hybrid graphics Prove $(1-a)(1-b)(1-c)(1-d)&gt;1-a-b-c-d$ and $ a,b,c,d$ are real and between 0 and 1.    I can do this with $$(1-a)(1-b)&gt;1-a-b  \\ 1-a-b+ab&gt;1-a-b  \\ ab&gt;0  $$  But with $c$ and $d$, this becomes much more difficult. Can someone nudge me in the right direction? When we drop a ball, it bounces back to the spot where we dropped it, due to the reaction forces exerted on it by the ground. However, if a person falls down (say, if we push them), why don't they come back to their initial position where they started their fall?  According to Newton's 3rd law of motion, to every action there is always an equal but opposite reaction. If we take the example of ball then it comes back with the same force as it falls down. But in the case of a human body, this law seems not to be applicable. Why? Why is the de-facto standard sigmoid function, $\frac{1}{1+e^{-x}}$, so popular in (non-deep) neural-networks and logistic regression?   Why don't we use many of the other derivable functions, with faster computation time or slower decay (so vanishing gradient occurs less). Few examples are . One of my favorites with slow decay and fast calculation is $\frac{x}{1+|x|}$.  EDIT  The question is different to  as I'm only interested in the 'why' and only for the sigmoid. Markdown in chat fails for multi-line messages Let moderators navigate the review queues at will This post is an expansion of  For many graduated sites (eg Physics), moderators do not want to apply the unilateral close vote in all cases--it is better to let some community votes pile on first.  We only use the unilateral in obvious cases then. This renders the CV queue useless to us. If we &quot;skip&quot; something, we can't get back to it. If we say &quot;do not close&quot;, it's removed from the queue and our community can't see it. If we use &quot;close&quot;, we are using a unilateral mod vote--we don't want to do that. What I propose is a set of &quot;next/previous&quot; buttons, that let us navigate the queues, looking for closeable questions, and coming back to the others later. Thoughts? Update and bounty reason Due to the recent Great Flag Overhaul, mods no longer see close flags in the flag queue. Which means that the only way to get to them is via the review queue 1 Shog's solution is ideal in this case; let mods view the close queue as a list. (In fact, a list similar to the old flag queue -- which shows a list of mods who have seen the post and what mod actions have already been taken -- would be great) On Physics, the majority of the flags that we get are VTC flags. And the community isn't always able to close things on their own, so we have to step in. However, powering through the close queue isn't an option; because &quot;skip&quot;ing a post2 leads to it being dumped into the void (I can't get back to it unless I have a link). Extremely inconvenient. Could the workflow here be improved? 1. The /tools listing for posts with outstanding close votes doesn't work because it doesn't list posts which have been flagged but not VTCd 2. which I may want to take a closer look at later; gotta be careful with the binding votes. Eclipse issues warnings when a serialVersionUID is missing.       The serializable class Foo does not declare a static final   serialVersionUID field of type long   What is serialVersionUID and why is it important?  Please show an example where missing serialVersionUID will cause a problem. Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group Problem: Show that the group given by the presentation $$\langle x,y,z  \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.   I have tried all sorts of manners to try to show that the relations given by the presentation above imply that $x=y=z=e$. However, I am stuck and would appreciate any hints as to how I should move forward. How many sequences of $n$ tosses of a coin that do not contain two consecutive heads have tails as the first toss? If you toss a coin $n$ times, there are $2^n$ possible sequences of heads and tails. Let $E_n$ be the set of sequences which do NOT contain two consecutive heads and $e_n$, the number of sequences in $E_n$.  Thus $E_3$ $=$ $\{$ $TTT$, $TTH$, $THT$, $HTT$, $HTH$ $\}$ and $e_3$ $=$ $3$.  How many elements of $E_n$ have $T$ as the first toss?  Please can anyone help me out here? Whenever I do camera tracking in Blender the reconstruction is always zooming out or not on the axis. I have a great quality camera, but the end results of the camera tracking always has the model sliding. If you have a solution please tell me. What is some good free software for doing physics calculations?  I'm mainly interested in symbolic computation (something like Mathematica, but free). Should I use the datetime or timestamp data type in MySQL? Would you recommend using a  or a  field, and why (using MySQL)?   I'm working with PHP on the server side.