Please point out what is wrong with this calculation: $i^2= (i) (i) $ $=\sqrt{-1} \sqrt{-1} $ $= (-1)^{1/2} (-1)^{1/2} $ $=((-1)(-1))^{1/2}$ $=1^{1/2}$ $=1$
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
What's the 'Frequent' sort function? What is the function of the  sorting option on the Questions tab? Solidify thickness doesn't correspond to specified scale I'm Having trouble with setting the thickness applied by solidify. The selected block is 15 by 28 cm (single mesh sides) and the thickness of solidify is set to 3mm (pointing inwards). Checking off even thickness makes it even worse. I was using the function in a different object earlier today and it worked just fine. Is there something I'm missing or is this just a bug?   Feel free to treat me like an idiot, I've literally started using blender yesterday :) Repeating an equation including its content and number Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$ Let $[z_1,z_2,z_3,z_4]$ denote the  of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized circle if and only if $[z_1,z_2,z_3,z_4]\in\mathbb{R}$.    I saw this statement claimed in , and wondered how one would go about showing it. I know of some facts we could work with:   First, the definition: A  in $\mathbb{C}$ is either a circle or a line in $\mathbb{C}$. I know that a Mobius transformation maps generalized circles to generalized circles. I know that if $f$ is a Mobius transformation and $z_j'=f(z_j)$, $j=1,2,3,4$, then $[z_1',z_2',z_3',z_4']=[z_1,z_2,z_3,z_4]$.   I have posted a potential proof as an answer below. Please let me know if you think it's complete. Thank you! Recovering Home folder after failure to boot I have a small issue. I was having some issues in getting some libraries to work and to resolve it, i deleted some shared libraries by mistake(sudo ...).Now ubuntu fails to boot up but i can access the files from recovery mode. I can see my home folder intact but i cannot even run sudo nautilus as it says it shared libraries not found. The data in my home folder is quite important and i want retrieving it is important. I am running a windows dual boot, if it helps. If anyone could help copy the home folder to a windows drive from the terminal itself(as i can access it at recovery mode) it would be of immense help. Thanks.:-) Is there a value that can be assigned to my time, i.e. an hourly rate, so that I can be given a receipt to remit to CRA come Income Tax Filing time?  I live in Canada. Which damage dice exactly does the Great Weapon Fighting fighting style allow you to reroll? Embedding DLLs in a compiled executable random access file in java I have the following fields:   Inventory control (16 byte record)   Product ID code (int – 4 bytes) Quantity in stock (int – 4 bytes) Price (double – 8 bytes)    How do I create a fixed length random access file using the above lengths? I tried some examples online, but I either get an EOF exception or random address values when I try to access them.  I tried some more examples and couldn't understand the concept very well. I'm trying a project with it and will try to explore more on it.  Here is some example data. There might be holes in the data where No. in stock could be 23 == 023.            Quantity ID. No.   In Stock   Price  -------   --------   ------  1001       476      $28.35  1002       240      $32.56  1003       517      $51.27  1004       284      $23.75  1005       165      $32.25   Thanks for the help. I am trying to evaluate the following:  $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a&gt;1$ and $a \in \mathbb{R}$  Any help will be much appreciated. Suppose there are several journals with the same impact factor. Is it better to submit my papers to one of them or each time I finished a manuscript it is better to submit to an equally good journal that I have not published in yet.  I am early in my career and I already published two papers in the same journal and I am going to finish my third paper, which potentially can be published in the same journal as my previous two papers. What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$? For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. The question is motivated from the simple exercise to determining whether the series $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$ is convergent. One may immediately get that it is convergent by the ratio test. So here is my question:     What's the value of $$\sum_{k=1}^{\infty}\frac{k^2}{k!}?$$ Did not receive site association bonus (+100 rep) $k$ times's draw from $n$ numbers with replacement, each number at least appear once Sample $k$ times with order from $n$ distinct numbers with replacement where $k\ge n$. Here "with order" means the sample is treated like a sequence rather than a set, so results like "1,3,3" and "3,1,3" are different.  The question is what is the probability that every number is sampled at least once? My answer is $1-\frac{{\sum\limits_{i = 1}^n {C_n^i\times i! \times {{(n - i)}^k}} }}{{{n^k}}}$ where $C_n^i$ denotes $n$ chooses $i$.  ${C_n^i\times i! \times {{(n - i)}^k}}$ means first choose $i$ ($i=1,2,...,n-1$) of numbers not to appear in the sample, and then we sample from the remaining $n-i$ numbers for $k$ times.   My question is, is there more elegant solution where the numerator can be directly computed? i.e. can we directly calculate the number of possible sample sequences in which every number has at least one occurrence? Thank you!    PS: my attempt was $C_k^n\times n!\times n^{n-k}$, meaning that we first choose $n$ positions in the sequence of length $k$ to place $n$ numbers, and then permute the $n$ numbers. This is to ensure every number to appear once. The remaining $n-k$ are sampled from the $n$ numbers without restriction.   However, I later found this seems not correct, and it counts the same sequence multiple times.  PS2: as pointed out in the comments, my first answer "1- ..." is not correct either. On my system (Windows Vista Home Premium 64bit) Python files are associated with Python interpreter:  C:\Users\Piotr&gt;assoc .py .py=Python.File C:\Users\Piotr&gt;ftype Python.File Python.File=c:\python\2.7\python.exe "%1" %*   Nevertheless when I run any Python script from command line a window titled Open With pops up asking me to Choose the program you want to use to open this file. Let $f$ be an entire function with $|f(z)|\le 100\log|z|,\forall |z|\ge 2,f(i)=2i, \text{ Then} f(1)=?$  I have no idea how to solve this one!  $g(z)={f(z)\over \log|z|}$ Then Can I say $g$ is constant by Liouville Theorem? I'm looking for this really cool music video that involved monsters and kids The video starts off with two kids a boy and a girl—both are speaking Spanish. They wake up and find that not only are their parents gone but also everyone in their town, so they pack up and try to find people.  I know there's a scene where they destroy a car.  When they're traveling they come across a big furry Totoro-like monster in a cave, and the monster is cool. Then it shows us that there's a bunch of kids and they're all running around having a blast and there's also a bunch of other alien looking monsters, but they must be good 'cause they're not attacking anyone.  And that's all I really remember. I've looked everywhere for it and if you can help it would be awesome.  I watched the video 5 or 6 years ago at least. I don't remember any females singing; I'm pretty sure it was a band of guys. It wasn't heavy metal or anything; it was more like pop. I don't remember much about the actual sound of it to be honest. I am tired of always trying to guess, if I should escape special characters like '()[]{}|' etc. when using many implementations of regexps.  It is different with, for example, Python, sed, grep, awk, Perl, rename, Apache, find and so on. Is there any rule set which tells when I should, and when I should not, escape special characters? Does it depend on the regexp type, like PCRE, POSIX or extended regexps? durbinWatsonTest() in R for non time-series data I am trying to understand whether the Durbin-Watson test is meaningful at all when applied to regression data that have no temporal order (eg. blood pressure ~ bodyMassIndex + exercise).   I would say no, as the autocorrelation should obviously vary with the order of the subjects (which is random), but I wonder whether the durbinWatsonTest in R involves a bootstrapping of the data, where order of the bootstrapping data is shuffled each time, and then perhaps an average autocorrelation is computed over the bootstrapping samples (but then again, shouldn't this average autocorrelation be zero for randomly sampled data?).  On the other hand, the value of the statistics seems to be quite stable over multiple runs of the function, so I am even more confused...  I came across it in Andy Field's book "Discovering Statistics with R", where it is used is an example with non-time series data, although in another, more recent book of the same author ("Adventures in Statistics") it does say that the test is applicable only to time series data. Although the merits of stepwise model selection has been discussed previously, it is becoming unclear to me what exactly is "stepwise model selection" or "stepwise regression". I thought I understood it, but not so sure anymore.  My understanding is that these two terms are synonymous (at least in a regression context), and that they refer to the selection of the best set of predictor variables in an "optimal" or "best" model, given the data. (You can find the Wikipedia page , and another potentially useful overview .)    Based on several previous threads (for example here: ), it appears that stepwise model selection is considered a cardinal sin. And yet, it seems to be used all the time, including by what seem to be well respected statisticians. Or am I mixing up the terminology?  My main questions are:   By "stepwise model selection" or "stepwise regression", do we mean: A) doing sequential hypothesis testing such as likelihood ratio tests or looking at p-values? (There is a related post here: ) Is this what is meant by it and why it is bad? Or B) do we also consider selection based on AIC (or similar information criterion) to be equally bad? From the answer at , it appears that this too is criticized. On the other hand, Whittingham et al. (2006; )1 seems to suggest that variable selection based on information-theoretic (IT) approach is different from stepwise selection (and seems to be a valid approach)...?  And this is the source of all my confusion.  To follow up, if AIC based selection does fall under "stepwise" and is considered inappropriate, then here are additional questions: If this approach is wrong, why is it taught in textbooks, university courses, etc.? Is all that plain wrong? What are good alternatives for selecting which variables should remain in the model? I have come across recommendations to use cross-validation and training-test datasets, and LASSO.   I think everyone can agree that indiscriminately throwing all possible variables into a model and then doing stepwise selection is problematic. Of course, some sane judgement should guide what goes in initially. But what if we already start with a limited number of possible predictor variables based on some (say biological) knowledge, and all these predictors may well be explaining our response? Would this approach of model selection still be flawed? I also acknowledge that selection of the "best" model might not be appropriate if AIC values among different models are very similar (and multi-model inference may be applied in such cases). But is the underlying issue of using AIC-based stepwise selection still problematic?  If we are looking to see which variables seem to explain the response and in what way, why is  this approach wrong, since we know "all models are wrong, but some are useful"?   1. Whittingham, M.J., Stephens, P.A., Bradbury, R.B., &amp; Freckleton, R.P. (2006). Why do we still use stepwise modelling in ecology and behaviour? Journal of Animal Ecology, 75, pp. 1182–1189.