Summation $\sum_{j=2}^{n-1}j^2$ Properties
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
Can you claim a Double Sacrifice Bonus? Two very closely related questions:  If a card has two different "Sacrifice this creature for X" abilities on it, does sacrificing it give both benefits?  Likewise, if a card requires a creature to be sacrificed, but that creature has a "Sacrifice this creature for X" ability, do you get the sacrifice benefit when you sacrifice it for the other requirement? The problem is this:      Suppose $I \subseteq R$ is a nilpotent ideal and there is $r \in R$ with $r \equiv r^2 \pmod I$. Show $r \equiv e \pmod I$ for some $e \in R$ idempotent.   I have spent a few hours rolling around in abstracta with no destination. I believe that if I could write down a concrete example of this, the example could guide me through the abstract definitions and show me where to look for an idempotent in $R$. I have looked at 2x2 matrices and could not find any such examples. Might anybody have one? How do I reference an alias in a WHERE clause? I flashed new firmware and before doing so backed up apps and data with Titanium Backup. How do I restore? I have the folder on my PC which contains two .gz files and a properties file. Ksplice is an amazing software (or whatever it is). It changes the way kernel updates work: Instead of rebooting after an update, it patches memory, so you don't need to do anything. jQuery ui tabs An excited electron loses energy in the form of radiation. The radiation constitutes photons which move at a speed $c$. But is the process of conversion of the energy of the electron into the kinetic energy of the photon instantaneous? Is there a a simple way to visualize this process rather than math? I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).  I've already constructed two non-isomorphic total orderings. They are $$ 1 &lt; 2 &lt; 3 &lt;\ldots\\ 1 &gt; 2 &gt; 3 &gt;\ldots $$ where "$&lt;$" and "$&gt;$" are  strict total ordering relations.  Thanks in advance for your help. The draft feature was discussed a lot  and in other posts, but I didn't see much talk about the OP's mention of being able to have multiple.     Optionally, allow multiple draft questions and a page to edit/delete them.    I think the draft feature where it autosaves and stores it server-side is great, but in most things where they have a draft feature (like email clients), they allow you to have multiple drafts (also stored server-side).  The expiration can even work the same as it does for the single draft system right now.  You wouldn't even need full page to store them, I think the user profile page could have a tab for it like it currently has for answers, questions, and everything else.    It could look very similar to how questions look, with it's title and tags listed there. I've stumbled upon a strange behaviour with Windows machines, which seems to be fairly consistent between all Windows versions from Vista/2008 to 8.1/2012 R2; it doesn't happen instead when using Windows XP or Windows Server 2003.  The problem is this: when the network adapter is configured for DHCP and the DHCP server doesn't register DNS records on behalf of its clients (because it can't, or because it's not configured to do so), then the forward A record gets registered, but the reverse PTR record doesn't.  Some more details:   Both the forward and the reverse DNS zones are AD-integrated and accept dynamic updates. All computers are joined to the domain. All computers use the correct internal DNS servers, both when configured statically and when getting their configuration from DHCP. "Register this connection's addresses in DNS" is enabled in the network adapters. Everything is fine when a computer has a static IP address; both the forward and the reverse records get automatically registered. When the same computer is configured for DHCP, the forward record is registered, but the reverse record isn't. This happens for all computers with an OS version >= 6.0, and it's definitely not related to a single machine. No amount of ipconfig /registerdns will change anything. No errors are logged anywhere.   Why does this happen, and how can it be fixed?  And no, configuring the DHCP server to perform DNS registration is not an option here. Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i &lt; j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to express $\Sigma$ as a polynomial in the elementary symmetric functions, for $n=2,3,4$. Case $n=2$ is trivial: $\Sigma(X_1,X_2)= \sigma_1 = X_1+X_2$. Case $n=3$ is easy. I write $\Sigma$ as follows: \begin{align*} \Sigma(X_1,X_2,X_3) = (\sigma_1 - X_1)(\sigma_1 - X_2)(\sigma_1 - X_3), \end{align*} then apply  formally, obtaining \begin{align*} \Sigma(X_1,X_2,X_3) = \sigma_1^3 - \sigma_1 \sigma_1^2 + \sigma_2 \sigma_1 - \sigma_3 = \sigma_2 \sigma_1 - \sigma_3. \end{align*} The case $n=4$ seems harder to me, and I'm unable to work it out. I hope there is some trick to avoid unnecessary computations. My attempt is to write down $\Sigma$ completely: \begin{align*} \Sigma = (X_1+X_2)(X_3+X_4)(X_1+X_4)(X_2+X_3)(X_2+X_4)(X_1+X_3) \end{align*} and then multiply factors in groups of two, obtaining: \begin{align*} \Sigma = (\sigma_2 - X_1X_2-X_3X_4)(\sigma_2 - X_1X_4 - X_2X_3)(\sigma_2 - X_2X_4-X_1X_3), \end{align*} but it doesn't seem to help much. How can I solve the exercise? Why is gradient the direction of steepest ascent? How can I rewire half switched receptacles? In my bedroom the bottom half of all receptacles are controlled by one switch. Is it possible to have just one receptacle with top and bottom hot all the time?  The top half is hot all the time the bottom half is controlled by one switch.  Keep in mind there are six other receptacles in this room that are the same way controlled by the same switch. When I took the receptacle out the right side tab had been removed.  The top right screw had a black wire, the bottom right had a red wire, the top and bottom left screws had separate white wires.  I changed the receptacle out as suggested, I capped off the red wire and I capped off the white wire that was attached to the bottom left screw. I attached the other white wire to the top left screw and attached the black wire to the top right screw. I now have none of the receptacles working at all. What did I do wrong? I hope this isn't a duplicate.  Similar questions that I found are over a year old.  I have seen cases where shortly after an upvote (perhaps on an old answer?), the summary in my user activity dropdown shows inconsistent counts:    Particularly, the Slid a div offscreen using jQuery answer shows +40, when it was only updated once yesterday.  Now, if this is a monthly total for this question, then it is correct, but when I refreshed later I got:    Now it shows +10, and other entries completely disappear.  So, what is this supposed to show me?  At the very least I am confused about what I am supposed to see here.  Here is another example:    This shows +20 for the same question as before, after I received a single upvote last night.  What gives?  You can see from my reputation that there was only a sinle upvote recently: How do I enumerate the properties of a JavaScript object? How do I enumerate the properties of a JavaScript object?  I actually want to list all the defined variables and their values, but I've learned that defining a variable actually creates a property of the window object. how to write in TeX "computer zero" with slanted vertical line on it? In TeX, how is computing zero sign written with slanted vertical line on it? For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that  $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$    My initial approach is  Base Case: $(a_1^2+b_1^2) = a_1^2+b_1^2$ which is true. (Although it is trivial)  Prove the statement is true when $n=2$: We have $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2$$ (Thanks André Nicolas for pointing it out)   So if $a,b,c,d$ are integers, $ac,bd,ad,bc$ are all integers and integers are closed under addition and subtraction. Hence $(ac-bd),(ad+bc)$ are integers.   Inductive Hypothesis: $\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$ is true  Inductive Step: $$\prod_{i=1}^{n+1}(a_i^2+b_i^2) = \prod_{i=1}^{n}(a_i^2+b_i^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2)$$  Where $c$ and $d$ are integers.   But when we apply $n=2$, we have  $(c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (e^2 + f^2)$ where $e$ and $f$ are integers.  Hence, by the principle of induction, the statement we needed to prove is true. Where can I find a standalone installer for Windows 7 Service Pack 1? Why is the (non-relativistic) stress tensor linear and symmetric? From :     "[...] the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. [...] The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric."   What is the more detailed explanation for this, and/or where can I find it? understanding "mount" as a concept in the OS I want to understand what mounting is. It is used in different contexts and situations and I can't find resources which:   Describe the mount concept Explain the actions taken by the computer/OS/utility when a mount is performed How and in which situations mount is used Which features in the Linux mount command are of frequent use and some examples   ( I hear mount applied to diverse entities directories, flash drives, network card, etc )