can anybody please tell me what's happening here ?  $$1^2=1$$  $$2^2=2+2$$  $$3^2=3+3+3$$  $$x^2 = x+x+\cdots+x \mbox{  ($x$ times)}$$  differentiating both the sides   $$2x = 1 + 1 + \cdots+1 \mbox{  ($x$ times)}$$  thank u
Consider the following:   $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$   Therefore,   $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$   Take the derivative of lhs and rhs and we get:   $\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}} = 2x$   Which simplifies to:   $x = 2x$   and hence    $1 = 2$.     Clearly something is wrong but I am unable pinpoint my mistake.
What use are Steam Trading Cards? Is it purely a 'gotta-have-them-all' mindset, or do they actually serve some purpose?  I notice that you are able to sell them on a community market. For a test I listed one, and someone bought it for 50p. Not exactly a huge amount, but I don't understand why someone would spend real money on them.  Am I missing something?   Are they redeemable for games or money off vouchers or something 'tangible' that I'm missing? So I have a mac computer but i plugged in a windows keyboard in my mac. I am still trying to figure out how to paste commands into a command block. So how can I paste a command? How to delete duplicate LiDAR points? How can I delete duplicate LiDAR points? I incorrectly read the same las twice and saved as a one file resulting in duplicate XY points.  How should I delete these duplicate points? How to add video to iphone simulator I see lots of tutorials telling me how to root my phone (Motorola Droid), but none of them mention any potential "side effects."  Aside from a few relatively minor issues that seem to require rooting to fix, I'm basically happy with the phone and therefore a little leery of messing with a device that cost me a lot of money and is only a few months old.  Assuming that I'm just talking about rooting, and none of the typical post-root actions (like installing a custom ROM), am I taking any risks by choosing to root?   Does it have the potential to brick the device? Will it void my warranty? Will I stop getting updates? Will the cops show up at my front door the next day? (Yes, I'm being facetious.) Anything else I should be aware of?   (See also: ) Can the notion of vector space or algebra over a field be meaningfully extended to fractional dimensions, so that for example $\mathbb{R}^{-2/3}$ makes sense? Has this been explored somewhere?  I know that super vector spaces can be thought of as one way of generalizing vector spaces to negative integer dimensions. Is there a similar concept for dimensions that are rational numbers? I'm not talking about Hausdorff dimension, because it doesn't allow for negative rationals, and I'm rather looking for extensions from a more algebraic point of view (dimension as the trace of the identity map), without recurring to a given metric. Cycles in SHA-256 Let's say I start with a particular 256 bit value. Call this $v$. I then hash that value, and get another 256 bit value. Call this $\text{SHA256}(v)$. I take this value and get another 256 bit value. Call this $\text{SHA256}^2(v)$. More generally, let's call the result of hashing $v$ repeatedly $n$ times $\text{SHA256}^n(v)$.  Now my question is, how big will $n$ be, such that $\text{SHA256}^n(v) = v$?  It would seem to me, that if it's a giant sort of permutation, $n$ would have to be $2^{256}$, is that correct? Is that provable, or is there any information on this? (Just curiosity, really.)  Another question I had was, do all 256 bit strings have unique SHA256 values, and is there a way to show that? (Or, stated differently, can it be shown that there are no SHA-256 collisions in the language of all "256 bit strings"?) What's the easiest way to run GUI apps on Windows Subsystem for Linux as of 2018? I searched around, and currently there are two methods suggested; installing an enhancement for Windows Subsystem for Linux and installing an XServer.  I want to know which method is the most hassle-free (easy to install AND to use), and which one is less memory-heavy.  I just want Synaptic and CMake. Why couldn't that be a builtin feature? Based on this question: , LVS may be the right solution for my problem.   But when I went to its homepage , I found that LVS has been updated since Nov, 2008. The world's moving fast, and I don't know if LVS was obsolete or not.  Is LVS standing still, or there're some better solutions to replace it?  Thank you so much. How to make the measuring grid larger? When starting the blender, there is always a cube on the floor. The floor is actually a grid. My question is how to make this grid bigger so that we can measure big stuff? I am new to Stack  Exchange and I think it's very impressive.  One thing bothers me is that because it is so far down the line, I don't think beginners stand a chance of growing.  As time goes on, more and more questions are being answered and therefore if someone asks something with a really basic answer you can just comment. However, beginners can't comment until they have a reputation of 50. The only way they can build up is by answering questions. But if they answer a basic question long winded, they will be voted down because not so much effort was needed or because its already a duplicate and will lose the bit of reputation they have earned.  Can you tell me the future for beginners? I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following:  Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set $Y=\{(t,t^2,t^3)\;|\;t \in k\}$. Show that $Y$ is an affine variety.  The rest of the problem I have completed. I'm just not sure how to show that this set is closed and irreducible. I assume that they intend for $A^3$ to have the Zariski topology. Need clarification on the Draw 4 rule. - Player 1 lays down a Draw 4 Card and says the color is Red. - Player 2 does not have a Red card but has a Draw 4 Card. Can Player 2 lay down his Draw 4 Card and have Player 3 draw 8 cards? Duality discrete math problem This is the only answer I got wrong on my HW and the prof does not want to give us the correct answers before our midterm     The dual of a compound proposition that contains only the logical operators $\lor$ , $\land$ , and   $\neg$ is the compound proposition obtained by replacing each $\lor$ by $\land$ , each $\land$ by $\lor$ , each $\def\T{{\rm T}}\def\F{{\rm F}}$   $\T$ by $\F$ , and each $\F$ by $\T$ . The dual of $s$ is denoted by $s^*$. Find the dual of these compound propositions.      a) $p \lor\neg q$     I got $\neg p \land q$       b) $p \land (q \lor (r \land \T))$     My answer was $\neg p \lor (\neg q \land r)$       c) $(p \land \neg q) \lor (q \land \F)$     My answer was $(\neg p \lor q) \land  \neg q$  I have tried googling the problem and cannot come up with anything on duals and our lectures are online and upon reviewing do not see anything. I am just confused and looking for a little guidance on what was incorrect with my answers. Localisation is isomorphic to a quotient of polynomial ring I am having trouble with the following problem.      Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.    I just want a hint.  Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks I know I can only use the gym feature after hitting level 5, but are there any other privilege or rewards for the levels above 5?  For example, in Ingress, some items are only available for the higher level player. How about Pokemon Go? How to encrypt files using a cross-platform solution? In Oracle, I can re-create a view with a single statement, as shown here:  CREATE OR REPLACE VIEW MY_VIEW AS SELECT SOME_FIELD FROM SOME_TABLE WHERE SOME_CONDITIONS   As the syntax implies, this will drop the old view and re-create it with whatever definition I've given.  Is there an equivalent in MSSQL (SQL Server 2005 or later) that will do the same thing? In the begining of "" episode (Star Trek Next Generation 6x16) an Yridian tells Worf, that he has a certain information,      that Worf's father is still alive (apparently not killed in ) and is held in a prison camp on Carraya IV.   Yridian is an information broker and agrees to reveal details (i.e. location) only for a price.      Worf agrees.   How or with what Worf would pay for this? There is no economic factor inside the Federation (no money is being used) and I don't recall anything, Worf would have on board Enterprise, that would have enough value for an information this kind. So i was reading about  on Wiki and I found a interesting thing written there .    It says that the S. radius of the universe is as big as the size of the universe?   How is this possible?  Since most the universe is empty space shouldn't the S. radius of our universe be significantly smaller then 13.7 light years?