Strange artifacts along window borders after waking computer from sleep mode
Colored borders around windows after update
Knowing $|z|$ and $|w|$ (and possibly $z\overline{w}$), what are $|z+w|^2$, $|zw|^2$, $|z-w|^2$, $|z/w|^2$? Question 1: Let $z$ and $w$ be complex numbers satisfying $|z| = 4$ and $|w| = 2$. Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I plugged in $4$ and $2$ to each equation and got $36$, $64$, $4$, $4$, for question one. It was wrong.  Question 2: Let $z$ and $w$ be complex numbers satisfying $|z| = 5, |w| = 2,$ and $z\overline{w} = 6+8i.$ Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I don't know how to even start on question 2 because of the $z\overline{w}$. A similar question was asked , but it was specific to .NET 3.5. Specifically, I'm looking for the following:   What is the correct way to determine which .NET Framework versions and service packs are installed? Is there a list of registry keys that can be used? Are there any dependencies between Framework versions? How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? Am I too old for academia after a PhD? What is the maximum number of Locations Discovered that one can achieve in Skyrim? Can ASP.NET MVC generate elements with lower case name and id attributes The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is surjective (indeed bijective), but I'm afraid that I can't see why this is the case. I wonder whether you might be able to point me in the right direction?  Indeed, the proof begins like this:  "For each $n \in \mathbb{N}$, let $k_n, l_n$ be such that $n = 2^{k_n - 1} \left(2l_n - 1 \right)$; that is, $k_n - 1$ is the power of $2$ in the prime factorisation of $n$, and $2 l_n - 1$ is the (necessarily odd) number $\frac{n}{2^{k_n - 1}}$."  It then states that $n \mapsto \left(k_n , l_n \right)$ is a surjection from $\mathbb{N}$ to $\mathbb{N} \times \mathbb{N}$, and so ends the proof.  I can intuitively see why this should be a bijection, I think, but I'm not sure how to make these feelings rigorous?   I suppose I'd say that the map is surjective since given any $\left(k_n , l_n \right) \in \mathbb{N} \times \mathbb{N}$ we can simply take $n$ indeed to be equal to $2^{k_n - 1} \left(2l_n - 1 \right)$ and note that $k_n - 1 \geq 0$ and thus $2^{k_n - 1}$  is both greater or equal to one so is a natural number (making the obvious inductive argument, noting that multiplication on $\mathbb{N}$ is closed), and similarly that $2 l_n - 1 \geq 2\cdot 1 - 1 = 1$ and is also a natural number, and thus the product of these two, $n$ must also be a natural number. Is it just as simple as this?  I suppose my gut feeling in the proving that the map is injective would be to assume that $2^{k_n - 1} \left(2 l_n - 1 \right) = 2^{k_m - 1} \left(2 l_m - 1 \right)$ and then use the Fundamental Theorem of Arithmetic to conclude that $n = m$. Is this going along the right lines? The 'implicit' definition of the mapping has me a little confused about the approach.    On a related, but separate note, I am indeed aware that if $K$ and $L$ are any countable sets, then so is $K \times L$, so trivially, taking the identity mapping we see trivially that this map is bijective and therefore that $\mathbb{N}$ is certainly countable (!), and thus so is $\mathbb{N} \times \mathbb{N}$. Hence, it's not really the statement that I'm interested in, but rather the exciting excursion into number theory that the above alternative proof provides. Is there a place to dispute duplicate markings? I'm trying to get images in my document in two different ways:   On top of the entire page, centered On top of a single column   I tried it with both twocols and multicols. I ruled out twocols because it forces a new page every time you turn it on, because I need the abstract to be wide and the rest of the cover page to be two columns.  Here are some examples of both points (screenshot from a downloaded PDF):   I want to achieve both, but somehow I can only solve the first point with:  \end{multicols}  \begin{figure}   \centering     \includegraphics[width=0.5\textwidth]{figure}   \caption{Dataflow pipeline of the edge detector}   \label{fig:workflowedge} \end{figure}  \begin{multicols}{2}   Is it awkward to turn off multicols, then turn it back on again?  For my second point, I've tried everything, but almost always, the image doesn't show up at all! The only way I can get it to show at all is by specifying the H placement specifier, but that places the image inside the text where I defined it, not on top of the column.  This is my best try, but it hides the image and breaks all references to ??:  \begin{figure}[t!]   \centering     \includegraphics[width=0.25\textwidth]{figure}   \caption{Dataflow pipeline of the edge detector}   \label{fig:workflowedge} \end{figure}   and I'm getting the warning:  Package multicol Warning: Floats and marginpars not allowed inside `multicols' environment!.   How can I make a figure a non-float?  I also tried the figure* environment, but that just creates an image centered on top of the page, like my first point. Launch shell scripts from anywhere Is it possible to run my scripts located in ~/scripts from anywhere?  Say i want to run ./scriptname regardless of my current working directory. Auto-incrementing field in feature class using ArcGIS Desktop? So I'm relatively new to ArcGIS and ArcCatalog.  I have created a field name of PROJECTID in a feature class in a Personal Geodatabase.  The data type is TEXT.  This will be a ten digit ID.  The first five characters will be a acronym.  The sixth character will be a digit that represents the type of feature.  It will be a number.  The last four characters will be numbers that should be incremented starting at 0001.    Example:     WATER-1-0001  Screen shot of the field:   So my question is how can I do this in ArcCatalog.  I've seen some Python code that will autoincrement like the following:  rec=[OBJECTID] def autoIncrement(a):   global rec   return "water" + str(a) + "-" +  format(rec, '04d')   Again I'm a newbie when it comes to ArcGIS . Now I am getting an error NameError name 'OBJECTID' is not defined. Can not unlock my Android device as have no internet connection? Kepler problem in time: how do two gravitationally attracted particles move? Two particles with initial positions and velocities $r_1,v_1$ and $r_2,v_2$ are interacting by the inverse square law (with G=1), so that  $$ {d^2r_1\over dt^2} = - { m_2(r_1-r_2)\over |r_1-r_2|^3} $$ $$ {d^2r_2\over dt^2} = - { m_1(r_2-r_1)\over |r_1-r_2|^3} $$  (the inverse square law along the line of separation). What is the complete solution of these differential equations? What is the position of the two objects as a function of time?  After reading a lot on Wikipedia, I've come to the definition of center of mass and relative coordinates:   $$R(t) ~=~ \frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$$  $$\ddot{r}(t) r(t)^2 ~=~ (m_1+m_2)G$$  Where $R$ is the center of mass, and $r$ is the displacement between the particles... Is this correct? How do I proceed to solve the differential equation? Quotient group $SL(2,\mathbb{R})/SO(2,\mathbb{R})$ I am confused by expressions like $\mathbb{R}/\mathbb{Z}$, especially if I see something of the sort $SL(2,\mathbb{R})/SO(2,\mathbb{R})$. I don't know what to make of it. I understand what a quotient of a group for an equivalence looks like, but I can't get my head around expressions like above.   Is a quotient like this always a group and what do elements in the set $SL(2,\mathbb{R})/SO(2,\mathbb{R})$ look like? PrintWriter and PrintStream never throw IOExceptions Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java. I'm trying to understand a "subproof" of the divisor Master Theorem (Cormen et al., Introduction to Algorithms, page 99), and in some point they state:  $$\Large a^{\log_b n} = n^{\log_b a}$$  where $a\geq 1$, $b &gt; 1$ and $n = b^i$, for some $i$. That is, $n \in \{1, b, b^2, ...\}$  I would appreciate a simple explanation of this equality (just OK for a CS undergraduate). How to keep leading zeros in PHP integer Let $x, y$ and $z$ be real numbers such that $x^2+ y^2+ z^2= 1$. Find the   maximum and minimum values of $2x + 3y + z$ .       How can I able to solve the problem? Thanks for your help. PS4 Controller not connecting to PS4 and won't reset? I was playing my PS4 two days ago and then packed it up with two controllers that were both connected to this PS4.   I got to where I was going, set everything up and now the controllers won't connect. They show pulsing orange when they plug in, but when I hit the PS Button to connect, it blinks white 1-2 times and goes back to orange.   I have tried resetting them both.  Can anyone help?