A possibility of the theoretical derivation of Pauli's Exclusion Principle
Theoretical Proof of Pauli's Exclusion Principle
What are the options for traveling to the airports in Paris? Paris has 3 airports (Orly, Roissy, Beauvais), what are the options to go to or to come from there?  (There is also Vatry (XCR) but there is very few flights and i see the shuttle system is currently off.) I recently put a bounty on a question regarding the . I did this as the only two answers on that page are not satisfactory and I wish to encourage a tractable answer.  I am concerned that when the bounty period expires, if I do not select an answer for the bounty that it may be awarded automatically to one of these speculative and unsatisfactory answers. How might I prevent that? I could put an answer of my own stating "no satisfactory sources are available" and award the bounty to that, however I feel that there should be a better way.  I stress that I put the bounty on the question specifically because the current answers are not satisfactory and I do not want to award them. Why wasn't Restricted Section of the library better protected? Why does the standard error of the intercept increase the further $\bar x$ is from 0? The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^2}\right]$$ where $\bar{x}$ is the mean of the $x_i$'s.  From what I understand, the SE quantifies your uncertainty- for instance, in 95% of the samples, the interval $[\hat{\beta}_0-2SE,\hat{\beta}_0+2SE]$ will contain the true $\beta_0$. I fail to understand how the SE, a measure of uncertainty, increases with $\bar{x}$. If I simply shift my data, so that $\bar{x}=0$, my uncertainty goes down? That seems unreasonable.  An analogous interpretation is - in the uncentered version of my data, $\hat{\beta}_0$ corresponds to my prediction at $x=0$, while in the centered data, $\hat{\beta}_0$ corresponds to my prediction at $x=\bar{x}$. So does this then mean that my uncertainty about my prediction at $x=0$ is greater than my uncertainty about my prediction at $x=\bar{x}$? That seems unreasonable too, the error $\epsilon$ has the same variance for all values of $x$, so my uncertainty in my predicted values should be the same for all $x$.  There are gaps in my understanding I'm sure. Could somebody help me understand what's going on? Array#each vs. Array#map I'm running the following script on Ubuntu 14.04:  #!/bin/bash  apt-get purge -y nginx apt-get install -y nginx  date   When I run it like cat /tmp/script | bash, apt-get starts installing, then "date" is printed (not the actual date, but the command name), then the remaining apt-get output is printed.  If however I run the script like /tmp/script, it works as expected: printing the date after apt-get finished.  Why does this happen and how can I force bash to work when being piped to the same way it does when invoked directly? I masked an image in the UV/Image editor masking section. I then used that mask in the compositor for background replacement. However, the mask size seems to be different. Why is that? Here's a screenshot: Styling border to match fill in QGIS? In the past I have just manually matched the color of the border to the color of the fill, but this can be time consuming.   I am wondering, is there any way to have the program automatically match the border color to that of the fill?   The fill color is being assigned through the styling tab in layer properties.  Using QGIS 2.8.6 Boxplot interpretation: is it correct that a boxplot is missing a whisker? I have a data set that has customerID and daily visits. This dataset records the number of times customers visited the site in a day.   customerID   TotalVisits  1            6  2            17  3            20  4            25   Just to explore the data further, I tried plotting the TotalVisits values in a box plot. The plot doesn't seem to have a lower whisker. q1=1 and there are no values between q1 and  q1-1.5(IQR). The plot has q1, q2 and q3. The minimum value and q1 are the same.   Is this why the boxplot doesn't have the lower whisker? Could you please help me with the interpretation? Is it correct to draw a boxplot without any whiskers?  [I am new to the site and I am not able to post the image.] I need to knock out a quick animation in C#/Windows Forms for a Halloween display.  Just some 2D shapes moving about on a solid background.  Since this is just a quick one-off project I really don't want to install and learn an entire new set of tools for this.  (DirectX dev kits, Silverlight, Flash, etc..)  I also have to install this on multiple computers so anything beyond the basic .Net framework (2.0) would be a pain in the arse.  For tools I've got VS2k8, 25 years of development experience, a wheelbarrow, holocaust cloak, and about 2 days to knock this out.  I haven't done animation since using assembler on my Atari 130XE (hooray for page flipping and player/missile graphics!)  Advice?  Here's some of the things I'd like to know:   I can draw on any empty widget (like a panel) by fiddling with it's OnPaint handler, right?  That's how I'd draw a custom widget.  Is there a better technique than this? Is there a page-flipping technique for this kind of thing in Windows Forms?  I'm not looking for a high frame rate, just as little flicker/drawing as necessary.   Thanks.  Post Mortem Edit ... "a couple of coding days later"  Well, the project is done.  The links below came in handy although a couple of them were 404.  (I wish SO would allow more than one reply to be marked "correct").  The biggest problem I had to overcome was flickering, and a persistent bug when I tried to draw on the form directly.   Using the OnPaint event for the Form: bad idea.  I never got that to work; lots of mysterious errors (stack overflows, or ArgumentNullExceptions).  I wound up using a panel sized to fill the form and that worked fine. Using the OnPaint method is slow anyway.  Somewhere online I read that building the PaintEventArgs was slow, and they weren't kidding.  Lots of flickering went away when I abandoned this.  Skip the OnPaint/Invalidate() and just paint it yourself.  Setting all of the "double buffering" options on the form still left some flicker that had to be fixed.  (And I found conflicting docs that said "set them on the control" and "set them on the form".  Well controls don't have a .SetStyle() method.)  I haven't tested without them, so they might be doing something (this is the form):      this.SetStyle(ControlStyles.UserPaint, true);     this.SetStyle(ControlStyles.OptimizedDoubleBuffer, true);     this.SetStyle(ControlStyles.AllPaintingInWmPaint, true);    So the workhorse of the code wound up looking like (pf is the panel control):      void PaintPlayField()     {         Bitmap bufl = new Bitmap(pf.Width, pf.Height);         using (Graphics g = Graphics.FromImage(bufl))         {             g.FillRectangle(Brushes.Black, new Rectangle(0, 0, pf.Width, pf.Height));             DrawItems(g);             DrawMoreItems(g);             pf.CreateGraphics().DrawImageUnscaled(bufl, 0, 0);         }     }   And I just called PaintPlayField from the inside of my Timer loop.  No flicker at all. How do I find out the number, type, resolution &amp; refresh rate of additional monitors I can attach to my MacBook or MacBook Pro?  This is an attempt at a canonical QA, as per the Meta QA -   to try prevent the myriad "same but different" questions we get on every possible combination of computers, cable &amp; monitors There are perfectly good resources out there that Ask Different should not have to duplicate the efforts of. How many "polygons" does my model have? I have my model and in the info bar its show this   May I need to know how many polygons it has, but  there is no option to find polygons. Verts, Edge and Faces nothing polygon option there. In the website its say that the upload limit is 400,000 polygons. I did not get over that limit or how many polygons my model has? When is the product of $n$ subgroups a subgroup? R linear regression categorical variable "hidden" value In Spivak's Calculus 3rd Edition, there is an exercise to prove the following:  $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$  I can't seem to get the answer. Either I've gone wrong somewhere, I'm overlooking something, or both. Here's my (non) proof:  $$\begin{align*} x^n - y^n &amp;= (x - y)(x^{n-1} + x^{n-2}y +\cdots+ xy^{n-2} + y^{n-1}) \\           &amp;= x \cdot x^{n-1} + x \cdot x^{n-2} \cdot y + \cdots + x \cdot x \cdot y^{n-2} + x \cdot y^{n-1}\\ &amp;\qquad + (-y) \cdot x^{n-1} + (-y) \cdot x^{n-2} \cdot y + \cdots + (-y) \cdot x \cdot y^{n-2} + (-y) \cdot y^{n-1}\\           &amp;= x^n + x^{n-1} y + \cdots + x^2 y^{n-2} + x y^{n-1} - x^{n-1}y - y^2 x^{n-2}  - \cdots- x y^{n-1} - y^n \\           &amp;= x^n + x^2 y^{n-2} - x^{n-2} y^2 - y^n  \\           &amp;\neq x^n - y^n \end{align*}$$  Is there something I can do with $x^n + x^2 y^{n-2} - x^{n-2} y^2 - y^n$ that I'm not seeing, or did I make a mistake early on?  EDIT:  I should have pointed out that this exercise is meant to be done using nine of the twelve basic properties of numbers that Spivak outlines in his book:     Associate law for addition   Existence of an additive identity   Existence of additive inverses   Commutative law for additions   Associative law for multiplication   Existence of a multiplicative identity   Existence of multiplicative inverses   Commutative law for multiplication   Distibutive law How can I SSH with a remote server so I can browse files and edit them? I have a username and password to use. A few years ago Umber Ferrule wrote that the toolbars in the taskbar are defined in   HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\Streams.   This looks very plausible but cannot be the entire story.  Deleting this key will in no way affect the taskbar and on rebooting the system the key magically reappears.  Why do I want to know?  Simple.  I would like to export the key to a reg. file as a convenient backup. I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources.        One particular case is sentences in the form $a \rightarrow b$ , that is, material implications. While I fully understand the mathematical logic approach to the study of those kinds of formulas, sometimes I get confused about what someone ( using implication sentences in the ordinary language ) is trying to say.        In ordinary language, I often see affirmations in the form of an implication.  One example would be someone affirming that "the moon is made of cheese" $\rightarrow 2+2=4$. [ Kleene, Mathematical Logic ].      I perfectly understand that if we choose the interpretation that gives the antecedent as false ( and the consequent as true ) we get that the whole implication is also true, but I would like to understand EXACTLY what he is trying to say when he AFFIRMS a sentence in the form of an implication. Is he saying that under the commonly agreed interpretation of that sentence ( antecedent = false, consequent = true ), that implication becomes true?      Another example, totally different, would be someone affirming that $x \in \mathbb{N} \rightarrow x \in \mathbb{N}$ or $x \in \mathbb{Z}_{&lt;0}$. Is he now affirming the same kind of sentence or saying something totally different?      When he affirms that sentence, is he saying that he finds the sentence to be a tautology?   Why didn't he say that?          Then, with these two examples we have an affirmation of the same kind of sentence (implication) meaning completely different things, in one case he affirms to say that under the commonly agreed interpretation of the sentence, it is true. In another case, he affirms to say that there's no interpretation that makes the sentence false.       Am I missing something? There is really that ambiguity in ordinary language, when people are affirming sentences?  Can't I just take some general instance of the affirmation of a sentence in the form $a \rightarrow b$ in ordinary language and know exactly what he's trying to say, regardless of the propositions he chooses the substitute $a$ and $b$ for?       Thanks a bunch. Why don't elves have more babies? Negative R squared contradicts ssa/sst? I understand from this question -    that the R squared value of a linear regression model can be negative if the intercept is constrained. And this makes sense if you define R squared as -   $$R^2 = 1-\frac{SSE}{SST}$$  One says $SSE&gt;SST$. But then,  $$SST = SSA + SSE $$ Total Sum of squares  = Sum of squared errors + Sum of squared residual. And with this we get -  $$R^2 = \frac{SSA}{SST}$$ And now it is hard to imagine how $R^2$ can be negative. Aren't SSA and SST >0 always?