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Rings and some of their properties So I am studying for an exam and was wondering if anyone could explain some of the reasoning being the proofs of the following properties:   Let R be a ring with $a,b\in R$. Then    $a0=0a=0$  $a(-b)=(-a)b=-ab$ $(-a)(-b)=ab$   Proof 1:   $a0=a(0+0)=a0+a0$ hence $a0=0$   I don't really understand why $a0+a0$ proves that $a0=0$ Looks like flagging has been affected by releases today. Are these intended effects?   Higher daily limits (appear doubled for me, flagging of posts limit also doubled).  Explained by Shog9 below. Soft comment deletion  - e.g. for "accept rate" obsolete comments - it does in fact work, but the immediate feedback is gone.  This is fixed now. Has the throttling rate changed?  Seems possible to repeated flag without waiting 5 seconds.  If it was broken, it isn't anymore. Old soft deletions that were neither categorised as "helpful" or "declined" have all become "helpful".   If limits have been updated, can we get the new formula published? Not changed, as confirmed by Shog9. By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this  Edit: By using Hans Lundmark hint, I get:  $$(x^2+1)^2-x^2$$   Is it fully factorized? Please warn about exceeding 50 question limit prior to asking a new question How can I increase my Vault Dweller's Happiness? Is it true that $0.999999999\ldots=1$? Series prove or disprove statement Let $a_n$ be a sequence. $\lim_{n\to\infty} a_n =0 $ then the series: $\sum_{n=1}^\infty a_n, \ \sum_{n=1}^\infty (a_n+a_{n+1}) $ converge and diverge together.  It doesn't seem right, there are no convergence tests that fit this question yet every counterexample I try does not seem to work. Is this how I should see the sum? $\sum_{n=1}^\infty (a_n+a_{n+1})=2\sum_{n=1}^\infty a_n +a_{n+1}$ If so then there's no real difference between the two series since you multiply it by a constant and add just one more value. How are Contacts in Marketing Cloud counted? Salesforce bills Marketing Cloud on total number of Contacts within the system. I'd like to know what exactly is used to count these? Is this the Email Studio > All Subscribers list or the Contact Builder > All Contacts list?  When we have Marketing Cloud connector and synchronise the Contacts object, is this counted towards that number? (or only when they have been inserted into the All Subscribers / All Contacts list, as they are not done so automatically) Is the wave-particle duality a real duality? I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right? What's the past participle form of the word exit? Is it exit (irregular, like set)? exited? exitted? On one page I found exited but if that's the case why isn't it exitted (double t) like with the word emit - emitted? Is there a rule when the consonant at the end is doubled and when not? Triangular Numbers Modulo $k$ - Hit All Values? Of all the people we know, in the books, who were sent to Azkaban, the majority are Death Eaters (Dolohov, Lucius, Crouch Jr, Bellatrix and so on) or suspected to be involved with them (Sirius, Stan, et cetera). They were imprisoned for using the Unforgivable Curses, as well as for being accomplices of a seriously dark wizard. What other crimes are Azkaban-worthy, in normal circumstances? I'm looking for anything that's been stated in the books, or on Pottermore (basically, anything 'canon'), or anything one can conclude from information in the books. I think smaller crimes are punished by imposing a fine, but what about bigger ones? Surely murder and torture (i.e. crimes against humanity) aren't the only ones punishable by imprisonment? That certainly isn't the case in contemporary Muggle society. My thoughts Off the top of my head, I can name these crimes:  Being an unregistered Animagus1 That was what Hermione threatened Skeeter with.  Trespass and robbery2 That's how Sturgis Podmore was imprisoned.   Notes  Several of the arrests in the Half-Blood Prince and the Deathly Hallows were made by an authoritarian (if that's the right word on this case) government, in the time of war. They do not count as normal.  I don't mean that there should be an answer to this question, I was wondering if it was ever addressed in the books. I certainly don't expect a list of all people sent to Azkaban, and the reasons for doing so!     1 &quot;Well, yes,&quot; said Hermione calmly, taking a sip of her drink. &quot;Otherwise, as you very well know, I will inform the authorities that you are an unregistered Animagus. Of course, the Prophet might give you rather a lot for an insider's account of life in Azkaban.&quot;  The Beetle at Bay, Harry Potter and the Order of the Phoenix     2 Sturgis Podmore, 38, of number two, ... charged with trespass and robbery ... sentenced to six months in Azkaban.  Percy and Padfoot, Harry Potter and the Order of the Phoenix What software can I use to write on PDFs? I'm used to OS X where the system PDF viewer will let you draw and write text on top of any PDF document. You can then flatten it and save as a new PDF. It saves paper and time so you can fill in any form without printing it and scanning it back to the computer.  It also lets you insert a scanned image of your signature.  Is there software for Ubuntu that would do that? I am looking for an app that can monitor my C: (system) drive and find which program is creating a folder, with the rather generic name TEMP1\, on that drive.  I wouldn't mind if it could hook the creation of files too – as there a couple of files created as well (quite probably from another process). Google does not help me, as the names of those files are a bit generic too (not as TEMP1 though – dvmaccounts.ini and dvmexp.idx).   The problem is that every time I delete those, they are recreated on startup, so the utility I'm searching must start up on boot and also start fast. It should also be really clever, as my C: drive is huge. How can I read a file even when getting an "in use by another process" exception? Installing Kali on a USB drive - NOT Live USB or Persistence partition Greeting's, to clarify; my question is not creating a Live USB drive or a persistence partition but creating a USB with GRUB bootloader (or other) and Kali which can be used on any machine. This is similar to running other linux OS directly from USB. I am currently running Ubuntu from a pen drive. I installed the OS (Ubuntu) and the bootloader on the pen drive and I can boot off it anywhere. Here is a video explaining the same (video is not by me: )  Since, Kali's installer doesn't give option to select which drive to write the booloader to and directly writes on the HDD (sda1). This means even if I partition and install Kali on a pen drive since the bootloader will be installed on the machines fixed primary drive, the USB drive will fail to boot on a different device.  I have two questions:   Anyone has any idea how to write bootloader to a USB drive and what parameters should be given so that it boots with Kali installed on it. I am using graphic install mode for installing Kali on the USB drive, however there is no option for the install to be encrypted (Full disk encryption or at least /home folder to be encrypted). Does anyone know how can I install Kali with full disk / home folder encryption? Or enable is post install?   Thank You! Finite Abelian groups with the same number of elements for all orders are isomorphic Let $A$ and $B$ be finite abelian groups. Suppose that for every natural number $m$, the number of elements of order $m$ in $A$ is equal to the number of elements of order $m$ in $B$. Prove that $A$ and $B$ are isomorphic.  Idea Given that these groups are finite, I think you have to use the primary decomposition theorem somehow. How to use multiple HDRIs My question is, how would I make the visible HDRI/Enviroment Map be image 1 and then have reflections as Image 2? If someone can post a node setup, or a .py file that would be great &lt;3 I was thinking maybe using the Is Camera Ray node but I'm not confident it would work, and have no idea where to start. How do I set up my studio for shooting large rugs from the ceiling? Is the universe isomorphic to a universal turing machine? I often think about problems that require an understanding of the very essence of computation and its inherent limitations. So, my questions are as followed:   Is the universe isomorphic to a universal turing machine? Is the universe isomorphic to a universal quantum computer? What evidence do we have within our universe to support an answer to such a question? What corollaries follow from a universe that is or is not isomorphic to a universal turing machine (or universal quantum turing machine)? Is this question really answerable, and what is the current literature on this subject?   I'm mostly interested in the implications and corollaries that follow from a universe that is or is not isomorphic to a turing machine (or universal quantum computer) and the insights that it would provide us with, about the nature of computation.