Prove that there exists walks that each edge is in $G$
Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails
How do I see that every left ideal of a square matrix ring over a field is principal? As the question title suggests, how do I see that every left ideal of a square matrix ring over a field is principal? how to fix ioctl permissions denied error? I searched in lots of websites/forums but did not find a solution for this problem, and this site is last chance, this is problem: This problem have several other users too. my phone is Galaxy S7 Edge , exynos, Oreo 8.0 (Stock ROM), rooted with magisk latest version. root works well in other applications. In pentesting applications like csploit, Network Spoofer, WifiKill and others  root works well(grants permission)  but when start using options of app(for example, arp spoofing, network sniffer, MITM, etc) drops same error: [ERROR] Unable to retrieve local hardware address libnet ( ioctl: Permission denied ) what to do? Can this issue fixed? or exactly which permission is it?  Also one interesting thing, when I installed old version of WifiKill app it worked - killed wifi connection. Use Cauchy product to find a power series represenitation of   $$1 \over {(1-x)^3}$$ which is valid in the interval $(-1,1)$.   Is it right to use the product of $1 \over {1-x}$ and $1 \over {(1-x)^2}$ if I know thier expantion (HOW).. Is there another way? Pixel RGB with ImageMagick and Rails After plain installation of Ubuntu 16.04 LTS gnome-software doesn't work. It has opened once with GUI, but installation of Unity Tweak Tool didn't finish.  When starting again, GUI doesn't come up. When opening from terminal, a warning appears, but the application GUI doesn't open.  $ gnome-software (gnome-software:7052): Gs-WARNING: failed to open plugin /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so: /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so: cannot open shared object file: No such file or directory   Does anybody have same experience, and eventually, a solution?    According to the recommendation:  $ dpkg-query -S /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so gnome-software: /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so   Proposed reinstall of gnome-software did not help.    As workaround, use Synaptic Package manager and command line. Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$  then $f$ is linear.   I know this is a well-known result. Where can I find the proof ? Prove that $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$ If $\alpha + \beta + \gamma = \pi $ and $\tan(\frac{-\alpha + \beta + \gamma}4)\tan(\frac{\alpha - \beta + \gamma}4)\tan(\frac{\alpha + \beta - \gamma}4) = 1$  Then prove that:     $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$.   I have no idea how to go about this.   Please help. Origins of Tension First of all, I have a confusion about the definition and idea of Tension. For example, in my Physics Textbook, the idea of tension is written like this: &quot;Let's say there is a wire with a cross-section area of A and its length is $L_o$. If I hang the wire from a point and hang a weight of W on the other end of the wire, it will generate a repulsive force of T. This T is called the Tension Force.&quot; For me, this only explains to me what force will we call tension force, but it doesn't quite explain to me what is exactly this Tension force or where it originates from. Also, I have been doing pulley problems (Figuring out the acceleration of weights hanging from pulleys)  by first labeling all of the tension forces by $T_1$, $T_2$, and then doing some math with them to get my answer. But It bugs me that I am using these tension force values without even knowing where they came from or what they are. My idea was, maybe Tension force comes from the intermolecular bonds between the molecules of the rope, rubber or wire, etc. But even then, I have some different situational questions. For example:  Why you can't push a rope, copper wire against a wall?  What happens when I pull on two ends of a wire with 2 different forces?  Does the force increase proportionally to the applied force? Why does the wire break after a certain amount of force is exceeded.?  Does the idea of tension apply to everyday objects,(for example, a coffee mug, glass pane, etc). If it does, can I use something like a heat map or 3D equation to describe the tension in every point of the object? [Edit: more concisely, is there a mathematical way to show the various amounts of tension forces on a 2D (like a metal sheet) and 3D objects (like a football ) instead of a 1D object(like a string)].  If I take a metallic object like a paperclip or a thin wire. I can bend It multiple times and it eventually breaks into two pieces. I used to do this to pull of capacitors from old circuit boards. But does it happen with things like ropes or paper? If not, why it doesn't happen? ( Maybe it has something to do with intermolecular forces and the various internal stresses?)  Let's say I have an iron bar and I have thrown it out into space. Does that iron bar have equal stress distribution or still uneven or will the stress on all points become zero? (Is there even such thing as zero stress?) Can I safely uninstall games purchased from Steam? Can I uninstall purchased games from my computer and then later download them through Steam without having to re-purchase if I'd like to?   I'm running short on space on my laptop, looking to free up some space and wanted to be sure that I can safely re-download anything at a later time similar to how you can with the Xbox Live marketplace. TikZ: Draw angle with label between lines I need to find the intersection point of 2 line segments (lines are finite, i.e., they have end points).   e.g. segment 1 from $(x_1, y_1)$ to $(x_2, y_2)$ -- segment 2 from $(x_3, y_3)$ to $(x_4, y_4)$  you can assume $m_1$ and $m_2$ are the gradients of segment 1 and segment 2 respectively  similarly, $c_1$ and $c_2$ being the y-intercepts of segment 1 and segment 2 respectively  Using $y=mx+c$ I can easily find the equations of both lines and then derive the intersection point from those equations and check if that point actually lies on both the line segments (intersection point may not lie between the end points).  My problem is when I calculate the gradients $m_1$ and $m_2$ I do $\frac{y_2-y_1}{x_2-x_1}$ there for if one of the lines is vertical then I am going to have problems. How can I deal with this?  Another problem is that if the two segments intersect at a point which is also the same as one of the end points then I want to assume that they are not intersecting.  e.g. if segment 1 is from $(2, 3)$ to $(3, 7)$ and segment 2 is from $(3, 7)$ to $(7, 3)$ then I want to assume they don't intersect. What is dependency injection? How do I change the computer name? Sending `time` command into text file I want to get the result of time command into a text file but it's not working it only put blank space in the text file.  I already tried this commands,  A-  $ x=`time` $ echo $x &gt; log.txt $ cat log.txt  $   B-   $ time &gt; log.txt  real    0m0.000s user    0m0.000s sys     0m0.000s $ cat log.txt  $   C-  $ time &gt; log.txt 2&gt;&amp;1  real    0m0.000s user    0m0.000s sys     0m0.000s  $ cat log.txt  $    What I really want is this.   $ time &gt; log.txt $ cat log.txt  real    0m0.000s user    0m0.000s sys     0m0.000s Computing determinant without expansion $$\begin{align}\mathrm D &amp;= \left|\begin{matrix} (b+c)^2 &amp; a^2 &amp; a^2 \\ b^2 &amp; (a+c)^2 &amp; b^2 \\ c^2 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)\left|\begin{matrix} b+c - a &amp; a^2 &amp; a^2 \\ b - a -c &amp; (a+c)^2 &amp; b^2 \\ 0 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right| \\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ b - a -c &amp; a+c - b &amp; b^2 \\ 0 &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ 0 &amp; a+c - b &amp; b^2 \\ c - a-b &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\end{align}$$  Can $\rm D$ be further simplified without expanding ?  I feel it should be because this was competition question. Styling each feature of a layer individually I’ve a PostGIS table which is published as a Geoserver layer. Every feature in the table has some attributes that should style that feature (e.g color, size, opacity, external graphic url, ... etc).  I can create an SLD document and let it read its styling attributes from the table using &lt;PropertyName&gt;, but the problem is more complex than this, for example: some points features need to be styled as a simple circles, while some other need to be styled as external-graphic images. And from the SLD Reference, the SLD PointSymbolizer must have either &lt;Mark&gt; or &lt;ExternalGraphic&gt; element, for that reason I should create two rules one for the simple point symbolizer and the other for the external-graphic point symbolizer, and should add a filter to each rule so that it deals only with the appropriate features, something like:  &lt;FeatureTypeStyle&gt;     &lt;Rule&gt;         &lt;ogc:Filter&gt;             &lt;ogc:PropertyIsEqual&gt;                 &lt;ogc:PropertyName&gt;is-external-graphic&lt;/ogc:PropertyName&gt;                 &lt;ogc:Literal&gt;0&lt;/ogc:Literal&gt;             &lt;/ogc:PropertyIsEqual&gt;         &lt;/ogc:Filter&gt;         &lt;PointSymbolizer&gt;             &lt;Graphic&gt;                 &lt;Mark&gt;                     &lt;WellKnownName&gt;circle&lt;/WellKnownName&gt;                     &lt;Fill&gt;                         &lt;CssParameter name="fill"&gt;                             &lt;ogc:PropertyName&gt;fill-color&lt;/ogc:PropertyName&gt;                         &lt;/CssParameter&gt;                     &lt;/Fill&gt;                 &lt;/Mark&gt;                 &lt;Size&gt;6&lt;/Size&gt;             &lt;/Graphic&gt;         &lt;/PointSymbolizer&gt;     &lt;/Rule&gt;     &lt;Rule&gt;         &lt;ogc:Filter&gt;             &lt;ogc:PropertyIsEqual&gt;                 &lt;ogc:PropertyName&gt;is-external-graphic&lt;/ogc:PropertyName&gt;                 &lt;ogc:Literal&gt;1&lt;/ogc:Literal&gt;             &lt;/ogc:PropertyIsEqual&gt;         &lt;/ogc:Filter&gt;         &lt;PointSymbolizer&gt;             &lt;Graphic&gt;                 &lt;ExternalGraphic&gt;                     &lt;OnlineResource                xlink:type="simple"                xlink:href="smileyface.png" /&gt;                     &lt;Format&gt;image/png&lt;/Format&gt;                 &lt;/ExternalGraphic&gt;                 &lt;Size&gt;6&lt;/Size&gt;             &lt;/Graphic&gt;         &lt;/PointSymbolizer&gt;     &lt;/Rule&gt; &lt;/FeatureTypeStyle&gt;   My question is: Is there any better solution that is more dynamic? Some ships in Star Trek are referred to with definite articles. "The Enterprise", "The Defiant". Other ships are referred to without them. "Voyager", "Enterprise". Is there a pattern to or explanation for this behavior? Differences between ways to express future actions Deleted comments leave badge on iOS app When someone leaves a comment that involves you being notified, you get the notification both on the main toolbar on Stack Exchange and as a notification/badge on the iOS app.  If you read a comment on the website and cause the notification to be removed from the toolbar, the badge also gets removed from the iOS app.  However, when a comment that has previously caused a notification/badge to appear on iOS gets deleted, the badge remains on the iOS app while the notification gets removed from the website toolbar.  Could the badge also be removed from the iOS app in this situation? I want to make some changes to Windows that require using the Group Policy Editor (gpedit.msc).  Unfortunately the Group Policy Editor is not included with the Starter Edition, Home and Home Premium editions of Windows.  How can I install it?