Find all functions $f\colon \mathbb R \to \mathbb R$ such that $f(f(x)(y + 1)) = x[f(y) + 1]$, $\forall x, y \in \mathbb R$.
Functional equation: $f: \mathbb{R} \rightarrow \mathbb{R}$, $f((x + 1) f(y)) = y (f(x) + 1)$
We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$  EDIT: I see this has been dealt with before but at least with  I'm not making the fundamental mistake of assuming an incorrect definition of $i^2$. How can you strip non-ASCII characters from a string? (in C#) How can you strip non-ASCII characters from a string? (in C#) In the film and book series, Year 4 (Goblet of Fire), how is it Mad-Eye Moody/Barty Crouch could see Harry through his Magical Eye even when he was under the Invisibility Cloak?  As the cloak is a True Invisibility cloak, immune to charms and spells, and according to legend invisible to even Death, how did Mad-Eye detect Harry Potter under the cloak? How Nagini and Mrs. Norris were able to sense Harry were through other senses, such as smell and warmth, but not visually.  I'm referring to the scene/chapter where Snape follows Harry at night. Harry hides under the Invisibility Cloak but his foot is stuck at the broken staircase. Eventually Mad-Eye arrives, sees Harry through his Magical Eye, and talks Snape away. Binomial confidence interval estimation - why is it not symmetric? I have used the following r code to estimate the confidence intervals of a binomial proportion because I understand that that substitutes for a "power calculation" when designing receiver operating characteristic curve designs looking at detection of diseases in a population.  n is 150, and the disease, we believe, is 25% prevalent in the population. I have calculated the values for 75% sensitivity and 90% specificity (because that's what people seem to do).      binom.test(c(29,9), p=0.75, alternative=c("t"), conf.level=0.95)      binom.test(c(100, 12), p=0.90, alternative=c("t"), conf.level=0.95)   I have also visited this site:    Which is a java page which calculates binomial confidence intervals, and it gives the same answer.  Anyway, after that lengthy set-up, I want to ask why the confidence intervals are not symmetric, e.g. sensitivity is      95 percent confidence interval:    0.5975876 0.8855583      sample estimate probability: 0.7631579    Sorry if this is a basic question, but everywhere I look seems to suggest that they will be symmetric, and a colleague of mine seems to think they will be too. The three lengths \textwidth, \linewidth and \hsize seem to all hold the width of the current line. At least I never saw some real difference between \textwidth and \linewidth in my code. Both seem to be identical even when changed, e.g. by a minipage environment.  Could someone give me an elaborate explanation of the differences between these lengths? I like to understand them in detail. I know that the first two are from LaTeX and the latter one from plainTeX.  There is also \columnwidth which seems to be identical to \linewidth or \textwidth in one-column mode, right? Except when the line is reduced by a minipage or quote etc. environment. Infinite amount of flags I have been going through some old posts, flagging them as necessary. But I have noticed that the number of posts that I am able to flag is continually staying at ten. It seems that the new flagging system has messed this up.  I am able to keep flagging posts even after 10, so I know it isn't a UI bug. This is a very big problem, because I can flag infinitely many times. And I am pretty sure this isn't the intended behavior of the overlords This isn't because I am being awarded more flags. I flag more than 13 posts in 5 minutes, and I can keep going without restriction  Some of the flags were accepted, so now my number is up to 13. But it is still staying at 13 How does Windows know whether it has internet access or if a Wi-Fi connection requires in-browser authentication? According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.  Refer: Einstein's thought experiment of a Physicist in an Accelerated Box.  I find that the 'How' part of this question is somewhat easy to understand Mathematically but hard to visualize physically and on the 'Why' part I am totally blank. The title says it all.  If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?  So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?  How should I answer his question? Make Bash autocomplete guess the closest approximation I have a somewhat uncomfortable keyboard on my laptop, resulting in that I sometimes misspell things in Bash and when pressing Tab for an auto-completion nothing happens.   Is it possible to have Bash guess what I meant to type, by for example when wanting to change directory to Documents and then mistyping cd ocum[TAB], given no other folder beginning with "ocum" is present, Bash interprets it as cd Documents/ since it is the closest possible action to do?   Or is this idea terribly complicated to achieve? I'm trying to solve this problem from Hatcher 3.3.24.  Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, \mathbb{Z})$ is $\mathbb{Z^r}$ if $M$ is orientable or $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if $M$ is not orientable. In particular $r \geq 1$ if $M$ is not orientable. Using exercise 6 construct examples showing there are no other restrictions on the homology groups of closed 3-manifolds. (In the nonorientable case consider the manifold $N$ obtained from $S^2 \times I$ by identifying $S^2 \times \{0\}$ with $S^2 \times \{1\}$ via a reflection of $S^2$.)  I think I've solved the first part, here's my proof:   First note the following theorem: If $M$ is a closed connected and orientable n-manifold then $H_n(M, \mathbb{Z}) = \mathbb{Z}$ and $Tors(H_{n-1}(M, \mathbb{Z})) = 0$. If nonorientable, then $H_n(M, \mathbb{Z}) = 0$, $Tors(H_{n-1}(M, \mathbb{Z})) = \mathbb{Z_2}$.  Now note by Poincare duality and UCT we have that $H_2(M) \cong H^1(M) \cong \mathbb{Z^r} \oplus T_0 \cong \mathbb{Z^r}$.   In the nonorientable case consider the exact sequence:  $0 \longrightarrow H_2(M) \longrightarrow H_1(M) \longrightarrow H_0(M) \longrightarrow 0$  But we know $H_1(M) = \mathbb{Z^r} \oplus Z_2$ from the earlier theorem and $H_0(M) = \mathbb{Z}$.   Therefore $H_2(M) \cong ker(\mathbb{Z^r} \oplus Z_2 \rightarrow \mathbb{Z}) \cong \mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$.   For the rest of the question I'm quite lost. How did the house elves start serving wizards? How/Why/When did the house elves start serving wizards? I tried on my own to find the answer but I couldn't find anything relevant to my subject but I am bad at searching. Estimating Uniform distribution endpoints using data with errors Suppose I have a random variable $X$ ~ $Unif(0,\theta)$ where I want to estimate $\theta$. I draw a sample $X_1,...,X_n$.One way is to get a point estimate using e.g.  maximum likelihood estimation via the sufficient statistic $X_{(n)}$.  However, suppose I want a 95% confidence interval of sorts on the value of $\theta$, how should I construct one?  The reason why I am interested in confidence interval is because of the following issue, which is actually my main question (the above prelude on confidence interval is just a suggestion on possible approach). So actually my sample came with some noise, in a sense that I actually observe $X_1+E_1,...,X_n+E_n$ where $E_i$~$N(0,\sigma_i^2)$ independently from each other and from the $X$'s. How should I make some form of statistical estimation about $\theta$? I would prefer some form of confidence estimate. How to use PostgreSQL/PostGIS with ArcGIS for Desktop? I'm trying to find out why my application is very slow on a certain machine (runs fine everywhere else). I think i have traced the performance-problems to hard-disk reads and writes and i think it's simply the very slow disk.  What tool could i use to measure hd read and write performance under Windows 2003 in a non-destructive way (the partitions on the drives have to remain intact)? Can all real polynomials be factored into quadratic and linear factors? So I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens if the polynomial is higher degree and can't be factored. Later in the page, it says this:     However, it can be shown that any polynomial with real coefficients is a product of linear and/or irreducible quadratic factors with real coefficients.   And I was wondering, how do we know this and is this definitely true? Arrays in Java and how they are stored in memory Can I use a present-tense verb when writing about a past event? My sentence is     In Charles Dickens' book Oliver Twist, the stingy orphanage director doles out just one small bit of soup to the hungry orphans.   The main question is should it be doles or doled? Could it be either? Sets with no asymptotical density over $\mathbb N$ Let's consider the  defined by:      Take $ A\subset \mathbb N$; define the sequence $x_n= \dfrac{|A\cap[1,n]|}{n}$, and then if $\lim\limits_{n\to\infty} x_n$ exists, call it $d(A)$, the density of $A$ over $\mathbb N$.   My teacher said that the collection of sets that have a natural density is not an algebra, but I don't know why. I can't even find a set that does not have a natural density.  Please help me )= In high school we learned to say "than I" and "as I" because you could potentially add an "am" to the end of the sentence.  Examples:  "She is smarter than I."  (Think: "...than I am.") "He is as tall as I."  (Think "...as I am.")  So analogously, shouldn't it be "like I" as well:  "He is sincere, just like I."  (Think: "... like I am.")  But universally, it seems that we use "like me".  Where does this reasoning break down?  Is there history here?