A problem related to rank and nullity of matrices.
Finding rank-nullity of a given matices
Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$ I know that the improper integral  $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s&gt;0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can be obtained using the Residue Theorem). My question is to know whether it has a closed form for any non-integer $s &gt;0.5$. When to use "have" and "have got" Formatting floats differently based on placement Comment on Duplicate Reference: Why would this be marked duplicate when it was asked years prior to the question referenced as a duplicate?  I also believe the question, detail, and response is much better than the referenced question.  I've been a C++ programmer for quite a while but I'm new to Java and new to Eclipse.  I want to use the  to visualize some data I'm working with.  This code is organized like this:  ./com ./com/touchgraph ./com/touchgraph/graphlayout ./com/touchgraph/graphlayout/Edge.java ./com/touchgraph/graphlayout/GLPanel.java ./com/touchgraph/graphlayout/graphelements ./com/touchgraph/graphlayout/graphelements/GESUtils.java ./com/touchgraph/graphlayout/graphelements/GraphEltSet.java ./com/touchgraph/graphlayout/graphelements/ImmutableGraphEltSet.java ./com/touchgraph/graphlayout/graphelements/Locality.java ./com/touchgraph/graphlayout/graphelements/TGForEachEdge.java ./com/touchgraph/graphlayout/graphelements/TGForEachNode.java ./com/touchgraph/graphlayout/graphelements/TGForEachNodePair.java ./com/touchgraph/graphlayout/graphelements/TGNodeQueue.java ./com/touchgraph/graphlayout/graphelements/VisibleLocality.java ./com/touchgraph/graphlayout/GraphLayoutApplet.java ./com/touchgraph/graphlayout/GraphListener.java ./com/touchgraph/graphlayout/interaction ./com/touchgraph/graphlayout/interaction/DragAddUI.java ./com/touchgraph/graphlayout/interaction/DragMultiselectUI.java ./com/touchgraph/graphlayout/interaction/DragNodeUI.java ./com/touchgraph/graphlayout/interaction/GLEditUI.java ./com/touchgraph/graphlayout/interaction/GLNavigateUI.java ./com/touchgraph/graphlayout/interaction/HVRotateDragUI.java ./com/touchgraph/graphlayout/interaction/HVScroll.java ./com/touchgraph/graphlayout/interaction/HyperScroll.java ./com/touchgraph/graphlayout/interaction/LocalityScroll.java ./com/touchgraph/graphlayout/interaction/RotateScroll.java ./com/touchgraph/graphlayout/interaction/TGAbstractClickUI.java ./com/touchgraph/graphlayout/interaction/TGAbstractDragUI.java ./com/touchgraph/graphlayout/interaction/TGAbstractMouseMotionUI.java ./com/touchgraph/graphlayout/interaction/TGAbstractMousePausedUI.java ./com/touchgraph/graphlayout/interaction/TGSelfDeactivatingUI.java ./com/touchgraph/graphlayout/interaction/TGUIManager.java ./com/touchgraph/graphlayout/interaction/TGUserInterface.java ./com/touchgraph/graphlayout/interaction/ZoomScroll.java ./com/touchgraph/graphlayout/LocalityUtils.java ./com/touchgraph/graphlayout/Node.java ./com/touchgraph/graphlayout/TGAbstractLens.java ./com/touchgraph/graphlayout/TGException.java ./com/touchgraph/graphlayout/TGLayout.java ./com/touchgraph/graphlayout/TGLensSet.java ./com/touchgraph/graphlayout/TGPaintListener.java ./com/touchgraph/graphlayout/TGPanel.java ./com/touchgraph/graphlayout/TGPoint2D.java ./com/touchgraph/graphlayout/TGScrollPane.java ./TG-APACHE-LICENSE.txt ./TGGL ReleaseNotes.txt ./TGGraphLayout.html ./TGGraphLayout.jar   How do I add this project in Eclipse and get it compiling and running quickly? launch sms application with an intent Where can we find frequently updated details about travel/entry restrictions due to Coronavirus (nCoV-19)? Due to the outbreak of nCoV-19 (Novel Coronavirus) many Countries have imposed several entry restrictions. But these are quite confusing for a lot of people. Some countries allow transit through parts of China (like Shanghai) but some restrict transit even through Hong Kong/Macau. Philippines restricted passengers coming from Taiwan too, but I think that that was later changed.  Where can one find a list of such restrictions for all the countries, ideally a source that is updated frequently Do things use apostrophe for indicating possessive? Possible Duplicate:          If someone owns something I would say: Mom's car.  But if the owner is not a person, does it actually own it according to English rules or common usage? Which phrase is right?         The house windows.   The house's windows.   The windows of the house. (Or something like that.)      (Feel free to modify this question so it would be clear for other people.) I have a Xonar DX (CMI8788 driver) 7.1 surround sound card installed on my Ubuntu 11.10 system.   I want to use the outputs of the card as 4 separate stereo outputs and I am struggling with ALSA to get it to work.  This is my asound.conf:  #/etc/asound.conf pcm_slave.eightchannels {         pcm "hw:0,0"            # or "hw:1,0" for the second card         channels 8 } pcm.stereo1 {         type plug         slave.pcm {                 type dshare                 ipc_key 87882222                 slave eightchannels                 bindings [ 2 3 ]         } } pcm.stereo2 {         type plug         slave.pcm {                 type dshare                 ipc_key 87882222                 slave eightchannels                 bindings [ 4 5 ]         } } pcm.stereo3 {         type plug         slave.pcm {                 type dshare                 ipc_key 87882222                 slave eightchannels                 bindings [ 6 7 ]         } } pcm.stereo4 {         type plug         slave.pcm {                 type dshare                 ipc_key 87882222                 slave eightchannels                 bindings [ 0 1 ]         } }   This is my aplay -l output:  $ aplay -l **** List of PLAYBACK Hardware Devices **** card 0: DX [Xonar DX], device 0: Multichannel [Multichannel]   Subdevices: 1/1   Subdevice #0: subdevice #0 card 0: DX [Xonar DX], device 1: Digital [Digital]   Subdevices: 1/1   Subdevice #0: subdevice #0 card 1: Intel [HDA Intel], device 0: ALC662 rev1 Analog [ALC662 rev1 Analog]   Subdevices: 1/1   Subdevice #0: subdevice #0 card 1: Intel [HDA Intel], device 1: ALC662 rev1 Digital [ALC662 rev1 Digital]   Subdevices: 1/1   Subdevice #0: subdevice #0   This is my aplay -L output:   The system seems to completely ignore my configuration. However, I am sure that the config is used, because when i insert a random string at the top, i get a parsing error.  Any help is greatly appreciated. What counts as an attack? I was looking over the class feature of the 'fiend' patron warlock in 5e. I'm a bit confused on what actions the player needs to take before they can use the 'Hurl Through Hell' ability on an enemy. The PHB states:     Starting at 14th level, when you hit a creature with an attack, you can use this feature to instantly transport the target through the lower planes.   I'm not clear on what an 'attack' means in this context. Does it have to be a physical attack like a sword or crossbow? Do magical attacks like Eldritch Blast that require an attack roll count? What about spells with no attack roll? Deducing $\sin x + \cos x \ge 1$ from $(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$ So in trig, say I have an acute angle $X$. And one can intuitively conclude that $\sin x + \cos x \ge 1$, but how does the fact that $$(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$$ tell me that it is true that $\sin x + \cos x \ge 1$? I don't quite see the connection. Thanks How would you explain why $e^{i\pi}+1=0$ to a middle school student? Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it simply, you don't understand it well enough” so how would I explain it. Coincidence, purposeful definition, or something else in formulas for energy In the small amount of physics that I have learned thus far, there seems to be a (possibly superficial pattern) that I have been wondering about.  The formula for the kinetic energy of a moving particle is $\frac{1}{2}mv^2$.  The formula for kinetic rotational energy is $\frac{1}{2}I\omega^2$.  The formula for energy stored in a capacitor is $\frac{1}{2}C \Delta V^2$.  The formula for energy delivered to an inductor is $\frac{1}{2}LI^2$.  Finally, everyone is aware of Einstein's famous formula $e=mc^2$.  I realize there are other energy formulas (gravitational potential energy, for example) that do not take this form, but is there some underlying reason why the formulas above take a similar form? Is it a coincidence? Or is there a motivation for physicists and textbook authors to present these formulas the way they do? Isekai manga where the mc is reincarnated through his own magic The MC reincarnated himself through his own magic after his party defeated the demon king. Before he used his magic to do so, he took the daughter from the demon king and traveled with her over the world. After he used his reincarnation magic he was born in a noble family where the daughter of the demon king is now working as a maid in this family. He also has an older/younger (not sure) brother. Through his knowledge he starts his training of body, swordsmanship and magic as a baby. Years later his brother wants to play with him but he denies him every time. But one day he accepts it to play tag. His brother can't catch him. After that, he requests from the mc to teach him his magic. He denied again, but made an offer. Only when he can catch him, would he teach him his magic. I hope I'm not mixing up different mangas. I try to create a 2D game for Android. It looks like Cut the rope (it isn't a clone but it have same level representation) if it has a value. My problem that I can't understand that optimal sprite sizes I should use. I.e. what target screen resolution should I use to sprites didn't have anomalies on HD-devices (like Samsung Galaxy S7 and etc., phones and tablets)?    To make my question is clearer I ask about this 1 unit = X pixels for xxhdpi-screens. What is X?    P.S. Sorry if it's a duplicate I will be grateful for link on this post cuz I couldn't find it. Iridescence In Cycles? Anyone knows what software tool is used to draw these circuit schematics I was reading online and found this site:   I was very curious how this professor drew all these circuit diagrams in his lecture notes (such as this one: ). They just look very very good.   And example is shown below:      To be more specific about what I mean, by "publication quality" I mean the following:   The software must output vector graphics, such as eps, or windows meta files. It would be good to have control over the style of the graphics, such the style of the components, line width, figure size, etc. It would be good to have a graphical user interface. I understand LaTeX is very powerful but it's just too painful to write a 100 line script to generate a circuit diagram.   So in short, I'm not really looking for a schematic capture program. I'm looking for a program that can generate pretty circuit schematic illustrations so that I can put them in papers and presentations.  (I am aware of other posts on this forum, such as this one: .  None of the software mentioned in that post seems to get you publication quality figures.) Can you update to macOS 10.14 (Mojave) without converting to APFS? At WWDC 2018 macOS 10.14 or macOS Mojave was disclosed.  I am currently on macOS Sierra (10.12) and am looking forward to upgrading to macOS Mojave once its released.  I was wondering if it will be possible to update to macOS Mojave without converting to APFS, similar to  trick from OS X Daily on how to upgrade to macOS High Sierra (10.13) without converting to APFS with the command /Applications/Install\ macOS\ High\ Sierra.app/Contents/Resources/startosinstall --converttoapfs NO. pkexec command in a .desktop file I made a .desktop file for Androxyde's Flashtool (utility for Sony Xperia devices that I have to open with an executable file in its folder) that requires root privileges to use fastboot utilities. I used to make it work with gksu, but I'm on Ubuntu 15.04 and gksu is now old.  I tried to modify the exec line from Exec=gksu /home/natasha/FlashTool/FlashTool to Exec=pkexec /home/natasha/FlashTool/FlashTool Then, I read about pkexec doesn't allow to run X11 applications and so I override in this way:      The problem now is: It asks me the password but the Flashtool's GUI doesn't start. BUT if I execute that command in terminal, the program starts without problems. What can I do? Which Distribution Does the Data Point Belong to? Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}$\ $(-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$  My attempt :   Assume $f$ is strictly positive  (if not n=0 works). Let $Z(f^{(n)})$ the number of zeros of $f^{(n)}$ on $(-1,1)$. Then we have: $$ \forall n\in\mathbb{N}:\;f^{(n)}(-1)=f^{(n)}(1)=0 $$ so that iterating the Rolle theorem to the successive derivatives of f we have  $$  \;Z(f^{(k+1)}) \geq Z(f^{(k)}) +1 $$ thus, by induction : $$ \forall n\in\mathbb{N}: \;Z(f^{(n+1)}) \geq Z(f^{(n)}) +1 $$ Therefore, $$ Z(f^{(n)}) \geq n $$ How can I find an another zero ?   I'm still looking for an another proof which does not use the lemma ..,  Thank you in advance.