We have the unit interval $[0,1]$ and we want to find the probability of picking two random numbers $a,b$ from that interval with $|a-b|&gt;0.5$.  Must I investigate $[0,1]×[0,1]$?  I don't have the faintest idea of how to solve this. The problem is that $[0,1]$ has infinite numbers to pick from… so how to calculate a probability with infinitely many items in the sample space?  I would be really happy if somebody shed a light on this.
Two numbers are chosen randomly and independently from the unit interval $(0,1)$.  What is the probability that they differ by more than $\frac{1}{2}$? Hint: Think of the $2$ numbers chosen as coordinates in the Cartesian plane. Draw a square with vertices at $(0,0)$,$(1,0)$,$(1,1)$, and $(0,1)$. Allowable choices for the $2$ points lie below the line $y=x-\frac{1}{2}$ or the above $y=x+\frac{1}{2}$. Find the combined area of these regions.  I need help on this. I know what $y=x-\frac{1}{2}$ and $y=x+\frac{1}{2}$ looks like but I am not understanding the problem as a whole
Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$ Let $p$ be prime and $(\frac{-3}p)=1$, where $(\frac{-3}p)$ is Legendre symbol. Prove that $p$ is of the form $p=a^2+3b^2$.   My progress:  $(\frac{-3}p)=1 \Rightarrow$ $(\frac{-3}p)=(\frac{-1}p)(\frac{3}p)=(-1)^{\frac{p-1}2}(-1)^{\lfloor\frac{p+1}6\rfloor}=1 \Rightarrow$ $\frac{p-1}2+\lfloor\frac{p+1}6\rfloor=2k$  I'm stuck here. This is probably not the way to prove that.  Also tried this way:  $(\frac{-3}p)=1$, thus $-3\equiv x^2\pmod{p} \Rightarrow$ $p|x^2+3 \Rightarrow$ $x^2+3=p\cdot k$ stuck here too.  Any help would be appreciated. How do I limit the number of rows returned by an Oracle query after ordering? Is there a way to make an Oracle query behave like it contains a MySQL limit clause? In MySQL, I can do this: select *  from sometable order by name limit 20,10  to get the 21st to the 30th rows (skip the first 20, give the next 10). The rows are selected after the order by, so it really starts on the 20th name alphabetically. In Oracle, the only thing people mention is the rownum pseudo-column, but it is evaluated before order by, which means this: select *  from sometable where rownum &lt;= 10 order by name  will return a random set of ten rows ordered by name, which is not usually what I want. It also doesn't allow for specifying an offset. I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?  Secondly, the book wants me to use the fact that if $n$ is an integer that is not a perfect square, then $\sqrt n$ is irrational. This means that $\sqrt 6$ is irrational. How are we to use this fact? Can we reason as follows:  $\sqrt 6$ is irrational  $\Rightarrow \sqrt{2 \cdot 3}$ is irrational.  $\Rightarrow \sqrt 2 \cdot \sqrt 3$ is irrational  $\Rightarrow \sqrt 2$ or $\sqrt 3$ or both are irrational.  $\Rightarrow \sqrt 2 + \sqrt 3$ is irrational.  Is this way of reasoning correct? Separate long math text under sum symbol into different lines? I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines? Why is orbital angular momentum not 0 when spin and linear momentum are not collinear?   Why can it be 0 when spin and linear momentum are parallel?  Like in the example of a scalar field at rest decaying into an electron and a positron via interaction $\frac{1}{2}g\phi \bar{\Psi}\Psi $   If spin is along x axis and three-momenta is along z axis, decay rate will be 0 if two spins are opposite because orbital momentum is not 0 after decaying. How do I create a 50x3 cell array from a 50x3 matrix. Keep just getting a 1x1 cell array with a 50x3 submatrix? How to stop an animation in C# / WPF? I have something like this:   barProgress.BeginAnimation(RangeBase.ValueProperty, new DoubleAnimation(     barProgress.Value, dNextProgressValue,     new Duration(TimeSpan.FromSeconds(dDuration)));   Now, how would you stop that animation (the DoubleAnimation)? The reason I want to do this, is because I would like to start new animations (this seems to work, but it's hard to tell) and eventually stop the last animation... How do real and effective user IDs work? When a normal user wants to make change to the passwd file, the user will by setuid be given the effective user access. User becomes root temporarily and can edit passwd.  However you can only edit your password right, and not everybody else? However your effective user access is root. So how come you're not allowed to change other passwords beside yours?  When you run a program with setuid, what does it mean actually when effective user is root, but real user id is still your name? What is the mechanism by which magnetic fields do work? I've seen some conflicted answers to this question in texts and on the web, in the case of a dipole, for example.   Do magnetic fields do work directly, or is it their induced electric fields that do work? Can a Tourist from a VWP country carry (Rent/ Borrow) a gun in say Florida Looking to find out if it is legal to borrow or rent a gun from an individual(s) or company to use to either visit the various gun ranges or wildlife hunting in Florida.  I have a shotgun licence in the UK for hunting etc... but looking to do something similar in Florida without falling foul of any gun carrying laws in transit to any gun range or hunting area?  I know various friends in the US that would lend the guns while there but are they required to be with me while attending any event? Basically not looking to be tethered to any third party... I have a program which I am trying to run, however when I run it; it just complains that it can't find a particular file.  However I have no idea which folder it is trying to find this particular file in. I have a copy of the required file, I just need to know which folder to copy it too.  Is there any way to show in real time which files are being accessed or which files are trying and failing to be accessed?  I am using Ext4 filesystem if that helps.  Thanks Fitting a plane to a set of points in 3D using PCA I am trying to estimate a midplane of a 3D model using the midpoints of paired landmarks, in order to reconstruct missing data (midplane refers here to the middle/saggital plane of the cranium which cuts the skull into two symmetrical halves, left and right).  I therefore need to estimate a plane from 27 points in 3D. I need to get the equation of the plane as $$ax+by+cz=d.$$  I have looked into orthogonal regression and principal component analysis (PCA) as methods, however I didn't take maths past A-levels and am struggling. I know I can supposedly use the eigenvectors to get the equation of the plane of best fit, but need someone to explain exactly how. I am using R for the PCA but am not great at R either.  Alternatively, if there's a better way to estimate the plane I would be glad to hear it! "compare with" or "compare to" Please check both sentences and let me know, which one is correct and why?   She compares me with her boyfriend.  She compares me to her boyfriend. Money making in Runescape Merge two rows in SQL Auto revision summary has wrong plural: "added 1 characters in body" As seen at:           Update: I'm 101% sure the down votes were due to the lack of free hands.    Update 2: likes me, like me not, like me, likes me not... yeah ! For many of my datasets I have the same imported data duplicated numerous times, each iteration displaying the data values for a different column (in my case it's fish catch weight per gridded cell per year, so lots of iterations of the same grid, with a different pattern of values per year).  For presentation purposes, it strikes me as helpful to use the same scale for each year, so I go to the original data and divide the maximum value into (arbitrarily) 10 classes and manually input the values and labels after setting the desired colour ramp and input data field:   This is extremely tedious, not just the manual excel calculation part, but also the input interface for the values &amp; labels*, hence my intention to make use of the 'save style' option. HOWEVER: when I save the style as QML and load it for the next layer, it saves everything - including the data field (in my case, year). When I change this field, QGis automatically generates the values &amp; labels based on the data for that field, ditching all of the values &amp; labels I set.   Is there any way to avoid this happening &amp; thereby duplicate class number, values and labels for each field by simply loading a style? (somewhat meta) Am I going around this the wrong way, OR, is there a different face I could use to climb this mountain? Spider Identification Illinois I noticed a really huge, beautiful web on my porch last week, and tonight, I discovered its resident. I'm thinking it's an Araneus diadematus (Cross Orbweaver), but I was hoping for some confirmation.   Can anyone confirm this or offer a better suggestion? Salesforce Account as Entry source in Journey Builder Does NOT work Is there a printf converter to print in binary format? I can print with printf as a hex or octal number.  Is there a format tag to print as binary, or arbitrary base? I am running gcc. printf(&quot;%d %x %o\n&quot;, 10, 10, 10); //prints &quot;10 A 12\n&quot; print(&quot;%b\n&quot;, 10); // prints &quot;%b\n&quot;