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Question: For every polynomial $P(x)$ of degree at n we have $$\sum_{i=0}^{n+1}(-1)^i\binom{n+1}{i}P(i)=0$$  Well I was inducting on degree of the polynomial, for $n=0$ its true and I got a bit confused in the inductive step, actually I cant figure the inductive step.   Is this property (if true) a famous one?  Thank you for your help. Were there any canon examples of Harry being an innately powerful wizard? Was Harry Potter an innately powerful/skillful Wizard in any way, based on canon?  It seems that whenever Harry Potter performed some unusually potent bit of magic (potent as far as require raw power AND/OR skill), most of the time we find that it wasn’t innately Harry’s skill, but a combination of:   some extraneous magic forces/rules/powers circumstance great reaction time (which is obviously extraordinary from his Quidditch playing; and is likely a larger factor in his dueling than the strength of his magic) general smartness or (the part that Harry’s detractors, including Voldemort missed) Harry’s strength of character/personality   The only two counter-examples I can think of which show him as having extraordinary or at least obviously above average magical abilities are:   His skill at flying brooms - while a part of it is pure physical skill, at least part is magical. His (corporeal) Patronus charm   As far as examples of his unusual feats being due to reasons listed above:   Obviously, surviving Avada Kedavra as an infant - due to Lily’s sacrifice Parselmouth talent - proxied via Voldemort’s piece of soul. Ability to see the Sorcerer’s Stone in the Mirror of Erised - due to purity of his character, nothing magical. Obvious lack of talent at potion making throughout all of the books (sans cheating in HBP) His win over the Basilisk - due to Hermione’s research, character traits, and as a consequence of the last one, getting Fawkes and Gryffindor’s Sword’s help.  Just to be clear, before someone flames me - what he did was obviously super extraordinary, BUT IMHO he would have achieved the same result if he was turned into a Squib for the duration of Chamber of Secrets fight. TriWizard Tournament entry - wasn’t him, it was Barty Crouch TriWizard Tournament second challenge - a combination of character and outside help (someone told him about Gillyweed) TriWizard Tournament third challenge - Mostly, Barty Crouch’s cheating  Note that I’m omitting the first challenge, since that one was the aforementioned Broom flying skills. His survival in the cemetery after Voldemort’s return - again, a combination of superior strength of character and super powerful wandlore magic that had nothing to do with his skill as a Wizard. The whole “last Horcrux” thing when surviving the second Avada Kedavra from Voldemort His ownership of the Elder Wand in the very last duel with Voldemort.   So, to paraphrase my original question, was there any canon/JKR evidence that Harry was anything but a completely average/mediocre wizard in any way, shape, or form if you take into account only his own magical abilities and skills? I am reading George Gamow's book  and a certain assertion Gamow makes seems a little startling considering what I already know of the subject of infinite cardinality.     But the number of all geometrical points, though larger than the number of integer and fractional numbers, is not the largest one known to mathematicians. In fact it was found the variety of all possible curves, including those of the most unusual shapes, has a larger membership than the collection of all geometric points, and thus has to be described by the third number in the infinite sequence. [Gamow, 22]   He later does explicitly say the number of all possible curves is $\aleph_2$. But nowhere else can I find reference to this being true, nor does he provide any sort of reasoning to its validity.  My question is how can you prove the set of all possible curves is of cardinality $\aleph_2$ and not $\aleph_1$ or $\aleph_0$? Should I assign dwellers to the storage room? I've played Fallout Shelter for a while now and normally use dwellers with appropriate SPECIAL stats for each room (Strength in the reactor, Perception in the water supply, etc).  However, I have a question regarding Endurance. The only room which uses this is the storage room. It doesn't produce anything; it has no cooldown; it just does nothing...  Even if I just build and upgrade the warehouse, my stock capacity is increased. Is there any good reason for me to waste my dwellers by assigning them to storage rooms? Merging two incoming broadband lines for faster Internet This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, navigate to the  tag. This question and answer is owned by the community! If you can improve it and make it more useful, you're welcome to edit it. I am building a trigger, a unit test, or other Apex code. I'm working with sObject records, and trying to access related record information. For example, I might have a structure like this: Account a = new Account(Name = 'Test'); insert a;  Opportunity o = new Opportunity(Name = 'Donation', AccountId = a.Id); insert o;  System.assertEquals(o.Account.Name, 'Test', 'Correct Account associated'); // Assertion fails!  Or, in a trigger, I might be aiming for something like this: trigger ContactTrigger on Contact (after insert) {     for (Contact c : Trigger.new) {         if (c.Account.Name == 'Test') {             // Take some action here         }      } }  Why doesn't this work? Why am I getting null values, or NullPointerExceptions? Preventing LibreOffice from using smart quotes (“ ”) instead of (dumb) straight quotes? I use LibreOffice Calc to make lists of programs with different parameters to run on an hpc cluster. For each batch of commands I must do a find and replace on the smart quotes, as they cause errors in command line arguments. It would be nice to cut this step out of my work flow. Does anyone know how I might change the default punctuation used?  Edit: Apparently the title font transforms dumb quotes into smart quotes.  Smart quotes have a different symbol depending on if they are at the start or end (“ ”)  Dumb quotes use the same symbol for the start and the stop ( " ) I am trying to understand how to see the total size of my hard drive &amp; how much is left. It was recommended on various searches to use the command below   df -h Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.6M  1.6G   1% /run /dev/sda1       219G   29G  180G  14% / tmpfs           7.8G   18M  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup cgmfs           100K     0  100K   0% /run/cgmanager/fs tmpfs           1.6G   64K  1.6G   1% /run/user/1000   However I don't understand any of these directories. I mean what are they, are these all directories or partitions or what? And why do they appear under "mounted on"? How does that bit work. I assume the /dev/sda1 is my solid state drive? So now if I install applications, which one of these directories does my software get installed  into? I assume it just gets installs in the / directory (I haven't even got a clue what the / directory is). Can someone explain what the other directories are and what they are used for? Should I file Form 1040 or Form 1040 NR? I am currently on H1 B (since October 1, 2017). Last year when on F1 OPT, I had filed Form 1040 NR for my taxes. Which form is advisable for 2017, considering I am on H1 B now. What options do I have? What is the greatest integer that divides $p^4-1$ for every prime number $p$ greater than $5$? What is the greatest integer that divides $p^4-1$ for every prime number $p &gt; 5$?  This was on a practice math GRE so it's probably really easy. I need to cast single figures (1 to 9) to (01 to 09). I can think of a way but its big and ugly and cumbersome. I'm sure there must be some concise way. Any Suggestions How long does it take for DNS records to propagate? Search [tag1] OR [tag2] Is there a way to search questions that are tagged with either tag1 or tag2?  Actually, I would like the general filter,      ([tag1] AND [tag2], ...) OR ([tag3]   AND [tag4], ...) OR ...   And even     [tag1] AND ([tag2] OR [tag3])   I need that because I'm looking for interesting questions and I have to search for each tag separately. How can I make sure texts from friends with iPhones don't go to iMessage instead of my Android phone? I have always used an Android phone, but text messages from certain friends get hijacked by iMessage. How do I prevent iPhones and MacOS Messages from sending me iMessages? I need all messages to be SMS or MMS. Some friends with iPhones will text me, but I don't receive the message. Today I noticed the Message app on my Mac had notifications. I never open this app since I don't use iMessage, but I looked today, and found multiple &quot;lost&quot; messages from several weeks ago. Please note that this is NOT the same question as . My Android phone is not, and has never been, registered with iMessage, and I have already followed those instructions. Another poster () had the same question as me----but it was closed, being deemed as the same as the Removing phone number question. It's actually fairly different. I have tried:  Leaving my Messages app closed on my Mac. But somehow messages from these people get hijacked. Deregistering my mobile number using Apple's online Deregister iMessage tool. When I attempted this, I got the message &quot;Your phone number is not registered with iMessage.&quot; Finding a setting on my Android phone that addresses this. The problem is apparently (and logically seems to be) on the Apple end.  Things  says he tried:   Closing Messages.app. I still get macOS notifications of messages. In Messages &gt; Preferences… &gt; Accounts &gt; [my email address], untick &quot;Enable this account&quot;. Then my friends messages disappear and I never see them at all, at least until I reenable in Messages.app. Looking in System Preferences and on icloud.com and appleid.apple.com    to see if there is some setting I can disable—but I've not found    anything relevant. Using Apple's Deregister iMessage tool to unregister my mobile    number. When I attempted this it gave me the message &quot;Your phone number is not registered with iMessage.&quot; The energy of an electromagnetic wave The intensity of an electromagnetic wave is only related to its amplitude $E^2$ and not its frequency. A photon has the same wavelength as the wave that's carrying it, and its energy is $h  f$.   So if a laser wave is kept at the same amplitude and the wave length is reduced, why does its intensity remain the same even though its photons now carry less energy?  Why are the intensities of electromagnetic waves so different to sound waves (and other waves travelling through a medium) which are related to $f^2  E^2$? The book says:     Concentration. Some spells require you to maintain concentration in order to keep   their magic active. If you lose concentration, such a spell ends.   So what happens if, for example, you are already concentrating on a spell, and cast Witch Bolt without concentrating. Would you get the initial arc of blue energy, or would it not even go off?  I am asking if you can cast a spell that says concentration without concentrating for an immediate effect. This would be useful to avoid interrupting your existing concentration. In essence, if you are concentrating on thing A (doesn't have to be a spell since other things need concentration), can you cast a concentration spell getting an immediate effect without losing concentration on thing A?  Since the definition of concentration is to keep the spell active (meaning it is already active) I would think that you would get the initial effect (if it had one) without the need to concentrate on it. Why does HTML require that multiple spaces show up as a single space in the browser? How to type latin small letter c with cedilla I run Ubuntu on a DELL XPS 13 notebook that has an US keyboard configured as US international. After upgrading to version 13.10 I can no loger get a cedilla by typing '+C. Instead, I get a "ć".  What should I do to get a "ç"?  I am using Ubuntu Gnome 13.10 64 bit. Let $ \boldsymbol{x} $ be a vector of $n$ numbers in the range $ \left[0, c \right] $, where $ c $ is a positive real number.  What's is the maximum of the variance function of this $ n $ numbers? Maximum in the meaning what spread of the number will maximize the variance? What would be a tighter bound for other assumptions on the spread of the numbers.  The variance of the vector $ \boldsymbol{x} $ is given by:  $$ \operatorname{var} (\boldsymbol{x}) = \frac{1}{n} \sum_{i = 1}^{n} {\left( {x}_{i} - \overline{\mathbf{x}} \right )}^2 $$  Where the mean $\overline{\boldsymbol{x}}$ is given by:  $$ \overline{\boldsymbol{x}} = \frac{1}{n} \sum_{i = 1}^{n} {x}_{i} $$ Inequality from Challenge and Thrill of Pre-college Mathematics