If $f:A \rightarrow B$ is a bijection and $S \subseteq A$, prove that $f(A \backslash S) = B \backslash f(S)$
Are there some good overviews of basic facts about images and inverse images of sets under functions?
Are film capacitors affected by DC bias? After looking around, I have found quite a few things about how DC bias reduces effective capacitance for a ceramic, or electrolytic capacitor, however nothing seems to talk about the effect on film capacitors. I've found a couple of papers about film caps (such as: ), and they mention frequency and temperature, but I can't find anything about DC bias reducing capacitance.  I have obviously checked the data sheet of the component I plan on using (), but there isn't anything about DC bias in that (unless I've just failed at spotting it).  This suggests to me that they aren't affected by DC bias, but that seems a bit of a bold claim, and I can't see that stated anywhere.  Are film capacitor values affected by DC bias? If so, can anyone point me to some data about what the roll off is?  I am planning to use some film capacitors in an application with 48-60V DC across the capacitors and a ripple current around 30 amps. How are first digits of $\pi$ found? Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either multiplication or division or any other operation on numbers.  Then how are the first digits of $\pi$ found -     3.1415926535897932384626433832795028841971693993...   In fact, more than 100,000 digits of $\pi$ are found (sources - )  How is that possible? If these digits of $\pi$ are found, then it must be possible to compute $\pi$ with some operations. (I am aware of breaking of circle into infinite pieces method but that doesn't give accurate results.)  How are these digits of $\pi$ found accurately? Can it be possible for a square root of some number to be equal to $\pi$? Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable Let's consider the space $\mathcal{L}^p(X, \Sigma, \mu)$ of all functions $f\colon X \to \mathbb{R}$ (or $\mathbb{C}$) for which: $$ \int\limits_X|f|^p \mu(dx) &lt; \infty. $$ Here $X$ is a metric space, $\Sigma\subset 2^X$ — sigma-algebra on $X$ and $\mu$ — is the measure on $\Sigma$. As usual, functions that are equal almost everywhere considered to be equivalent, so technically we are dealing with classes $$[f]=\{g\in \mathcal{L}^\infty(X,\Sigma, \mu) \colon \mu \{x\in X\colon f(x)\neq g(x)\}=0\}.$$ This space is endowed with integral metric $$\rho_p(f,g)=\left(\int\limits_X|f-g|^p \mu(dx)\right)^{1/p}$$  The statement is that space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable, where $\forall A,B\in\Sigma$ $$\rho_\Delta(A,B)=\mu(A\Delta B)$$ Here, again, we consider $A\in \Sigma$ as a class $[A]=\{A'\in\Sigma\colon \mu(A\Delta A')=0\}$ to have the identity of indiscernibles.  How do one prove the statement? How do I put text over symbols? This is difficult for me to explain, but how do I put words over a symbol? For instance, I want to add words over an equal sign. Anyone know the command/syntax for that?  Also adding ^ doesn't cut it..., I want the text to appear ON TOP. How can I bypass Activation Lock? I bought a used iPhone 4.  The previous owner re-set the phone, but did not remove the device from their iCloud.  It is asking me for their iCloud information when I try to set-up the phone.  I don't have contact with the previous owner anymore.  Is there a way to remove the device from their iCloud so I can use a new one? Why can't I adopt children despite having a steading in Falkreath? I bought two pieces of land. One from Morthal, and one from Falkreath. Morthal was the first plot of land I purchased. I only finished half of it. I bought the plot of land in falkreath because I liked the nature better. I finished the house completely in Falkreath. Every time I try to adopt a child off the streets it keeps saying "I have a place but no room for you" Even though I have the bedrooms in my Falkreath house. Why can I not adopt a child and have him/her live at my house in Falkreath? Where does matter go after reaching singularity in uncharged black holes? I am a layman in physics and just read about black holes on the internet. I read that matter encounters geodesic incompleteness in the singularity in an uncharged black hole.  I heard an analogy of geodesic incompleteness as a straight line on a paper reach a hole on the paper, so it cannot continue. But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)? So, if matter reaches the singularity, is it possible too (to reach another dimension)?   I also heard that the matter is annihilated when reaching the singularity, does it mean it disappear from this world, and violates conservation of energy? "I use to", or "I used to" Which is the correct sentence, if there is a correct one?     I use to be a hitman.        I used to be a hitman.   I've read the second sentence recently in a book, but I was sure it should be I use to be a hitman. Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime. Cross Browser Flash Detection in Javascript Does anyone have an example of script that can work reliably well across IE/Firefox to detect if the browser is capable of displaying embedded flash content. I say reliably because I know its not possible 100% of the time. If it was possible to dig a hole that went from one side of the Earth to the other... I have start learning Selenium WebDriver, but i am confused between Java and Python. Which language should i choose? I do have basic ideas of Core Java but haven't wrote a single piece of code with python.  So please suggest me which is the better language for Selenium WebDriver? Deleting my own question DNS subdomain (child) NS records If we have example.com we would setup things something like this: $ORIGIN example.com. @  1D  IN  SOA ns1.example.com. hostmaster.example.com. (                   2002022401 ; serial                   3H ; refresh                   15 ; retry                   1w ; expire                   3h ; nxdomain ttl                  )           IN  NS     ns1.example.com.           IN  NS     ns2.example.com.  since in the parent zone .com. there is already NS record for the example.com. child zone and A glue record for the ns1.example.com. for what purpose do we state NS records again in the child zone? and also, do we need to state A record for the name servers in the child zone itself? What is the difference between encrypting some data vs signing some data (using RSA)?    Does it simply reverse the role of the public-private keys?    For example, I want to use my private key to generate messages so only I can possibly be the sender.  I want my public key to be used to read the messages and I do not care who reads them.  I want to be able to encrypt certain information and use it as a product-key for my software.  I only care that I am the only one who can generate these.  I would like to include my public key in my software to decrypt/read the signature of the key.  I do not care who can read the data in the key, I only care that I am the only verifiable one who can generate them.  Is signing useful in this scenario? Sometimes you spend a day or more solving a technical problem, and when you finally solve it, you may want to post it on your blog (if you have one) to share with the community.  However, most developers don't have a lot of time to update their blogs, and that is why personal blogs are not ranked very highly on Google/Bing.  So even if you describe a brilliant idea, chances are that, except a few of your friends, nobody will ever find it. This is a common reason why some developers start their blogs and soon give up; it is not interesting to write for the community if you have no feedback on your posts.  Unlike personal blogs, Stack Exchange sites are highly ranked in Google (I was amazed when I saw that a question asked on SO just an hour ago appeared 3rd in my Google search!), so if your question and answer are valuable for others, they will be rated highly and will be visible for other developers.  With this in mind, is it good practice to post a question and answer it yourself immediately when you want to share an idea with the community? Normality of dependent variable = normality of residuals? This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!).   The assumptions of general linear models (t-test, ANOVA, regression etc.) include the "assumption of normality", but I have found this is rarely described clearly.   I often come across statistics textbooks / manuals / etc. simply stating that the "assumption of normality" applies to each group (i.e., categorical X variables), and we should we examining departures from normality for each group.   Questions:     does the assumption refer to the values of Y or the residuals of Y?  for a particular group, is it possible to have a strongly non-normal distribution of Y values (e.g., skewed) BUT an approximately (or at least more normal) distribution of residuals of Y?   Other sources describe that the assumption pertains to the residuals of the model (in cases where there are groups, e.g. t-tests / ANOVA), and we should be examining departures of normality of these residuals (i.e., only one Q-Q plot/test to run).  does normality of residuals for the model imply normality of residuals for the groups? In other words, should we just examine the model residuals (contrary to instructions in many texts)?  To put this in a context, consider this hypothetical example:   I want to compare tree height (Y) between two populations (X).   In one population the distribution of Y is strongly right-skewed (i.e., most trees short, very few tall), while the other is virtually normal  Height is higher overall in the normally distributed population (suggesting there may be a 'real' difference). Transformation of the data does not substantially improve the distribution of the first    population.  Firstly, is it valid to compare the groups given the radically different height distributions? How do I approach the "assumption of normality" here? Recall height in one population is not normally distributed. Do I examine residuals for both populations separately OR residuals for the model (t-test)?     Please refer to questions by number in replies, experience has shown me people get lost or sidetracked easily (especially me!). Keep in mind I am not a statistician; though I have a reasonably conceptual (i.e., not technical!) understanding of statistics.  P.S., I have searched the archives and read the following threads which have not cemented my understanding: Beamer: Change only several bullets in the list I would like to change the bullets in the second itemize to arrows.  \begin{alertblock}{The Aim of this Research}      \begin{itemize}         \item Item 1         \item Item 2             \begin{itemize}                 \item Item 2-1                 \item Item 2-2             \end{itemize}     \end{itemize}  \end{alertblock}   Form above, I would like to make:  ・Item 1  ・Item 2    =&gt; Item 2-1  (change bullet to arrow)    =&gt; Item 2-2   I've tried inserting \renewcommand{\labelitemii}{$\rightarrow$} before and after the second \begin{itemize} and , but they don't work. Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$ Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$.   I was thinking among the lines of:  Since $a$ and $b$ are coprime numbers, $\gcd(a,b)=1$. Then there exist integers $x$ and $y$ such that, $ax+by=1$.   Then, $a=(1-by)/x$, $b=(1-ax)/y$  So if I write $2^a+1$ as:  $2^{(1-by)/x}+1$  Then can I say that the above expression is equivalent to 0 (mod 3)? Opening PDF generated by Python AddIn using Report (*.rlf) file automatcally? As a test I have:   Authored a test.mxd with one layer called Localities in a single data frame called Layers Authored a test.rlf using View | Reports > Create Report of ArcGIS 10.2.2 for Desktop Created a Python AddIn which creates the expected PDF file from the Report (*.rlf).   This is the code which all works to create the PDF report from the Localities features within the current extent:  import arcpy  import pythonaddins  class ReportForExtent(object):     def __init__(self):         self.enabled = True         self.checked = False      def onClick(self):         mxd = arcpy.mapping.MapDocument("CURRENT")         df = arcpy.mapping.ListDataFrames(mxd,"Layers")[0]         lyr = arcpy.mapping.ListLayers(mxd,"Localities",df)[0]         arcpy.mapping.ExportReport(lyr,r"C:\temp\test.rlf",r"C:\temp\test.pdf","EXTENT",extent=df.extent)         del mxd   Is there a way to have the Python AddIn not just create the PDF file but also open it in Adobe Acrobat Reader?