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How can I solve this quantum mechanical "paradox"? How to change the voltage polarity in an inverting amplifier? I want to implement a simple circuit which can map -10V  +10V voltage range to 0V 5V voltage range. Thus I'm planning to use two op-Amps, one for the slope and the other for the offset. So basically I'm planning to use an inverting amplifier and a summing amplifier consecutively.  But Inverting amplifiers multiplies the input voltage with a minus sign. I need a positive slope. How can I overcome this?  Also Could you gibe me any ideas for a cheap op-Amp models for this purpose? Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$. I'm looking for some guidance in solving this. I have read through my text multiple times trying to find something similar to this and I'm coming up with nothing. I'm not really sure where to start on solving this problem.  The question simply states     Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.   I'm very new to binomial coefficients, and extremely lost. How to work out sqft/m of land coverage? I am currently self teaching myself on qgis whilst volenteering on a project on a river catchment and your help is hugely appreciated!  I need to be able to work out the percentage of land cover or sqft/m coverage. I have all the land uses on as a vector layer over my river catchment.   Many thanks and let me know what other info i need to give as a newbie! If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints. C++ map access discards qualifiers (const) The following code says that passing the map as const into the operator[] method discards qualifiers: #include &lt;iostream&gt; #include &lt;map&gt; #include &lt;string&gt;  using namespace std;  class MapWrapper { public:     const int &amp;get_value(const int &amp;key) const {         return _map[key];     }  private:     map&lt;int, int&gt; _map; };  int main() {     MapWrapper mw;     cout &lt;&lt; mw.get_value(42) &lt;&lt; endl;     return 0; }  Is this because of the possible allocation that occurs on the map access? Can no functions with map accesses be declared const? MapWrapper.cpp:10: error: passing const std::map&lt;int, int, std::less&lt;int&gt;, std::allocator&lt;std::pair&lt;const int, int&gt; &gt; &gt; as this argument of  _Tp&amp; std::map&lt;_Key, _Tp, _Compare, _Alloc&gt;::operator[](const _Key&amp;)  [with _Key = int, _Tp = int, _Compare = std::less&lt;int&gt;,  _Alloc = std::allocator&lt;std::pair&lt;const int, int&gt; &gt;] discards qualifiers I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.  $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$  Perhaps someone can help me out I don't understand how to move forward from here: $$\sum\limits_{k=1}^{n+1}{\frac{1}{\sqrt{k}}+\frac{1}{\sqrt{n+1}}\ge \sqrt{n+1}}$$ proof and explanation would greatly be appreciated :) Thanks :)  EDIT sorry meant GE not = fixed :) Recently I have started playing with jQuery, and have been following a couple of tutorials. Now I feel slightly competent with using it (it's pretty easy), and I thought it would be cool if I were able to make a 'console' on my webpage (as in, you press the ` key like you do in  games, etc.), and then have it Ajax itself back to the server in-order to do stuff.  I originally thought the best way would be to just get the text inside the textarea, and then split it, or should I use the keyup event, convert the keycode returned to an ASCII character, append the character to a string and send the string to the server (then empty the string).  I couldn't find any information on getting text from a textarea, all I got was keyup information. Also, how can I convert the keycode returned to an ASCII character? On thanksgiving day a woman posted in a facebook group that her father had just passed away (left her money) and in good deed because it was thanksgiving she was offering to pay people’s debts. Well I inboxed her and she pretty much wants me to open up a bank account with First Tech Credit Union and give her my log in and password info. Ofcourse its sketchy!   So I questioned her and asked why she wanted that instead of her sending me the money directly from her bank to mine. She then responded because she wants to keep records of my debts. (She asked for proof and I sent her screenshots ONLY of my outstanding balances) I heard about the depositing checks scam so I questioned her about how she would deposit the funds and she said cash.  Her facebook page is 100% legit, I screenshotted her pics and family profiles etc. We came to an agreement that I would only accept her help if she sends me a picture of her ID before me sending this info.   She literally just wants to deposit the money to me, She isn’t asking me to send nobody money from that account or anything. Of course the account will be empty when I provide it to her. When she does deposit the money I am planning to immediately change the password and obviously stay watching my account activity while she deposits the funds.   She actually also provided pics of other accounts she funded. My question is, is this a scam? And if so how could she possibly scam me? Specially if i have her information and A pic of her ID. She is a christian lady and looks like she has a good heart, We talked about my struggles and she actually looked into why I needed to be assisted with the money. By no means was she pushy or anything. I want to believe that she just wants to help me but I am also too skeptical. Preservation of ideals in localized rings Refer to localization of rings in Lang's Algebra p. 108. In particular, let $A$ be a commutative ring, $S$ a subset of $A$ that is a submonoid of the multiplicative monoid structure of $A$, containing the identity. Then  $S^{-1}A$ is the set of elements $\frac{a}{s}$ with $a \in A$ and $s \in S$.  We have $\frac{a_1}{s_1}=\frac{a_2}{s_2}$ if there exists $\bar{s} \in S$ such that $\bar{s}(s_2a_1 - s_1 a_2)=0$. Now consider the injection of $A$ into $S^{-1}A$ given by $x \mapsto \frac{x}{1}$. This is a ring homomorphism. Let $J$ be an ideal of $A$. Ring homomorphisms preserve ideals and so the injection of $J$ into $S^{-1}A$, call it $J^{*}$, will be an ideal. My problem is that i don't see why for any $\frac{a}{s} \in S^{-1}A$ and any $\frac{j}{1} \in J^{*}$ we will have that $\frac{aj}{s}$ will be in $J^{*}$, i.e. why there must exist an element $\kappa \in J$ such that $\frac{aj}{s}=\frac{\kappa}{1}$. It's common in C++ to name member variables with some kind of prefix to denote the fact that they're member variables, rather than local variables or parameters. If you've come from an MFC background, you'll probably use m_foo. I've also seen myFoo occasionally.  C# (or possibly just .NET) seems to recommend using just an underscore, as in _foo. Is this allowed by the C++ standard? Can't install libglew-dev because libcheese and libclutter don't have the required versions I'm getting the following error message when attempting to install libglew-dev,  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  libcheese-gtk23 : Depends: libclutter-gtk-1.0-0 (&gt;= 0.91.8) but it is not going to be installed                    Depends: libcogl15 (&gt;= 1.15.8) but it is not going to be installed  libcheese7 : Depends: libclutter-gst-2.0-0 (&gt;= 0.10.0) but it is not going to be installed               Depends: gstreamer1.0-clutter but it is not going to be installed  libclutter-1.0-0 : Depends: libcogl-pango15 (&gt;= 1.15.8) but it is not going to be installed                     Depends: libcogl15 (&gt;= 1.15.8) but it is not going to be installed E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   All those packages are however apparently already installed.  I have tried all possible solutions in  post, without any luck. I have not added any PPAs. The version of ubuntu is 14.04.2.  I have looked at similar posts, and tried the following:  Running  dpkg --get-selections | grep hold   prints nothing.  sudo apt-get -oDebug::pkgProblemResolver=true install -f   Gives me nothing of interest, 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Restrict certain mods to certain worlds How are constraint forces represented in Lagrangian mechanics? Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Prove via induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2).  Not sure when to apply the induction hypothesis and how to get the result $1- \frac{1}{(n+2)!}$.  Please help! thanks guys, youre the greatest! What's the added value of SD line over regression line when examining association between 2 variables? I'm trying to incorporate different practices to use when exploring a new data set. Especially, how to examine the association between two variables.  Steps for example (not necessarily by order):   plot a y-by-x scatter plot of the raw data to see the relationship visually. compute summary statistics for each variable (mean and sd) compute correlation coefficient r draw the OLS regression line, compute its slope and intercept etc....   I've come across the "SD line" in Freedman's Statistics book, which is defined as:     "the line that goes through the point of averages and climbs at the rate of one vertical SD for each horizontal SD" Freedman, D., Pisani, R., &amp; Purves, R. (2007). Statistics (4th edn).   Since this book ("Statistics") is a canonical textbook, I consider its choice to discuss the SD line as an indication for the line's importance. However, a simple google search for the term "SD line" doesn't yield as many independent results. Most of them come directly from Freedman's book. This tells me it's not a central concept in bivariate analyses in general.  When comparing the SD with the OLS regression line, it seems like the regression line is more informative (than the SD line) for predicting y from x. Therefore, I'm wondering if bothering to plot the SD line has any benefit or added value that I would not already have when plotting the regression line.  Example using mtcars dataset, focusing on association between weight and mpg  data(mtcars)  ## calculate means mean_wt &lt;- mean(mtcars$wt) mean_mpg &lt;- mean(mtcars$mpg)  ## calculate standard deviations sd_wt &lt;- sd(mtcars$wt) sd_mpg &lt;- sd(mtcars$mpg)  ## scatter plot plot(x = mtcars$wt, y = mtcars$mpg)  ## add the "point of averages" points(mean_wt, mean_mpg, col = "red", cex = 1.5, pch = 16)  ## calculate the slope of the sd line slope &lt;- -1*sd_mpg/sd_wt  ## plot the sd line curve(expr = x*slope + (mean_mpg - slope*mean_wt), add = TRUE, col = 'blue', lwd = 2, type = "l", lty = 2)  ## plot the regression line model &lt;- lm(mpg ~ wt, data = mtcars) abline(model, col = "orange", lwd = 2)  ## legend legend("topright",        legend = c("Regression line", "SD line"),        col = c("orange", "blue"),        lty = c(1, 2),        lwd = c(2, 2))     Thus, my question: how can the SD line increase one's understanding about the relationship between two variables, in a way that's either adding or complementing on what the regression line already tells? Schengen visa validity start date My Chinese in-laws intend to visit us in one month from now. Their visa is valid from Nov 23rd. Would it be possible for them to take the airplane on November 22nd in China and arrive in Schengen area on November 23rd?  Or does the visa already have to be valid upon departure?  In our particular case, they still need to transfer in Beijing first. The actual flight to Schengen zone effectively only departs on the 23rd so I guess that should be completely fine then? Can ANOVA be significant when none of the pairwise t-tests is? Is it possible for one-way (with $N&gt;2$ groups, or "levels") ANOVA to report a significant difference when none of the $N(N-1)/2$ pairwise t-tests does?   In  @whuber wrote:      It is well known that a global ANOVA F test can detect a difference of means even in cases where no individual [unadjusted pairwise] t-test of any of the pairs of means will yield a significant result.   so apparently it is possible, but I do not understand how. When does it happen and what the intuition behind such a case would be? Maybe somebody can provide a simple toy example of such a situation?  Some further remarks:   The opposite is clearly possible: overall ANOVA can be non-significant while some of the pairwise t-tests erroneously report significant differences (i.e. those would be false positives). My question is about standard, non-adjusted for multiple comparisons t-tests. If adjusted tests are used (like e.g. Tukey's HSD procedure), then it is possible that none of them turns out to be significant even though the overall ANOVA is. This is covered here in several questions, e.g.  and . Update. My question originally referred to the usual two-sample pairwise t-tests. However, as @whuber pointed out in the comments, in the ANOVA context, t-tests are usually understood as post hoc contrasts using the ANOVA estimate of the within-group variance, pooled across all groups (which is not what happens in a two-sample t-test). So there are actually two different versions of my question, and the answer to both of them turns out to be positive. See below. What's up with the scraper sites? Letter O with vertical line: operator null