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Why is homology invariant under deformation retraction/homotopy equivalence? I'm assuming that the answer to the question in the title is a resounding yes. Since Baryonic matter and dark matter interact via gravitational forces.  If this is the case how is information not lost if, as is presumed, dark matter does not interact via the electromagnetic force? $d_1(x_1,y_1)$ and  $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about proving the triangle inequality for $d$ ? Thank you for your help. Contemporary written usage of "whom" in objective case Possible Duplicate:          I was writing a LinkedIn recommendation one day, and ended up pondering for a while which of these forms to use:     … is a great developer whom I always   found easy to work with.      … is a great developer who I always   found easy to work with.   Both are basically correct in contemporary English, right? But is one or the other preferable, and if so, why?   (In this case I went with the latter, as it seemed more common (Google)  and I wanted to avoid sounding unnecessarily “archaic”, although I’m not sure whether that would have been the case. Also note that I try to write in a “friendly professional” style instead of overly formal one. :-) Retrieving Maxwell's equations from the minimum action principle I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.  Starting with the action:  $$S = \int dt \int d^3 x \mathcal{L}_{\mathrm{EM}} = \int dt \int d^3 x \left( -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + j_\mu A^{\mu}\right).$$  In the absence of any sources $j_\mu = 0$ and $c=1$, which after pulling in the negative sign implies  $$S = \frac{1}{2}\int dt \int d^3 x  \,\,\left(E^2 - B^2\right)$$ The next step is to take $\vec{E} = -\vec{\nabla} \phi + \partial_t \vec{A}$ and $\vec{B} = \vec{\nabla} \times \vec{A}$. Letting $\phi = 0$ and directly inserting these into the above yields:  $$S = \frac{1}{2} \int dt \int d^3 x  \,\left( (\partial_t \vec{A})^2 - (\vec{\nabla} \times \vec{A})^2 \right) $$  Now considering $\vec{A} = \vec{A}_0 + \delta \vec{A}$ as some minimal change in our vector potential, with $\vec{A}_0$ gives a minimum for the action and putting this into the above expression we have: $$\delta S = \int dt \int d^3 x \left(\partial_0 \vec{A}_0 \partial_0 \delta \vec{A} + \left(\vec{\nabla} \times \vec{A}_0\right)\cdot\left(\vec{\nabla} \times \delta \vec{A} \right) \right)+ \mathcal{O}(A^2)\,\,\,\,\,\,\, (1)$$ where I have the term  $$\mathcal{O}(A^2) = \int dt \int d^3 x \left( (\partial_0 \vec{A}_0)^2 + (\partial_0 \delta \vec{A})^2 - (\vec{\nabla}\times\vec{A}_0)^2-(\vec{\nabla}\times \delta\vec{A})^2\right)$$  The next step is where I lose our author and go from "Oh ok I get this" to "completely clueless". He rewrites $\delta S$ as:  $$\delta S =  \int dt \int d^3 (\delta x) \vec{A} \mathcal{F(\vec{A}_0)}+ \mathcal{O}(A^2)$$  where $\mathcal{F} = \frac{\delta S}{\delta A}$. What is the purpose of $x\to \delta x$ in the above expression? Is the functional derivative $\mathcal{F}$ a vector quantity? He has it bolded with a bolded $\vec{A}$ in the above expression. One last question takes the above expression further. We assume $\delta \vec{A}$ vanishes at infinity, as it should, and integration by parts yields:  $$\delta S =  -\int dt \int d^3 x \left( \partial^2_0 \vec{A}_0 - (\vec{\nabla} \times \vec{\nabla})\times \vec{A}_0  \right)\delta \vec{A}\,\,\,\,\,\,\,\, (2)$$ How exactly did we go from (1) to (2)? Thanks I appreciate your time! What are acceptable reasons for flagging as "Requires Moderator attention"? After reading , I feel I don't quite understand this function. What are the guidelines for flagging a question as &quot;Requires Moderator attention&quot;? Simple animation using C#/Windows Forms 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 to install Windows 10 Home 19018 update? I'm using windows home edition, I have the latest update installed 1909.  I want to install docker for windows 10 home edition. It keeps saying  Docker Desktop requires Windows 10 Pro/Enterprise (15063+) or Windows 10 Home (19018+).  Should I join Windows insider? How long should I wait for that update to be released if I don't want to join windows insider? Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Has Stack Overflow been haiku-hacked? What is the best way to share/distribute Excel macro? Is there a way to know when a systemd timer will run next? I am testing a systemd timer and trying to override its default timeout, but without success. I'm wondering whether there is a way to ask systemd to tell us when the service is going to be run next.  Normal file (/lib/systemd/system/snapbackend.timer):  # Documentation available at: # https://www.freedesktop.org/software/systemd/man/systemd.timer.html  [Unit] Description=Run the snapbackend service once every 5 minutes.  [Timer] # You must have an OnBootSec (or OnStartupSec) otherwise it does not auto-start OnBootSec=5min OnUnitActiveSec=5min # The default accuracy is 1 minute. I'm not too sure that either way # will affect us. I am thinking that since our computers will be # permanently running, it probably won't be that inaccurate anyway. # See also: # http://stackoverflow.com/questions/39176514/is-it-correct-that-systemd-timer-accuracysec-parameter-make-the-ticks-slip #AccuracySec=1  [Install] WantedBy=timers.target  # vim: syntax=dosini   The override file (/etc/systemd/system/snapbackend.timer.d/override.conf):  # This file was auto-generated by snapmanager.cgi # Feel free to do additional modifications here as # snapmanager.cgi will be aware of them as expected. [Timer] OnUnitActiveSec=30min   I ran the following commands and the timer still ticks once every 5 minutes. Could there be a bug in systemd?  sudo systemctl stop snapbackend.timer sudo systemctl daemon-reload sudo systemctl start snapbackend.timer   So I was also wondering, how can I know when the timer will tick next? Because that would immediately tell me whether it's in 5 min. or 30 min. but from the systemctl status snapbackend.timer says nothing about that. Just wondering whether there is a command that would tell me the delay currently used.  For those interested, there is the service file too (/lib/systemd/system/snapbackend.service), although I would imagine that this should have no effect on the timer ticks...  # Documentation available at: # https://www.freedesktop.org/software/systemd/man/systemd.service.html  [Unit] Description=Snap! Websites snapbackend CRON daemon After=snapbase.service snapcommunicator.service snapfirewall.service snaplock.service snapdbproxy.service  [Service] # See also the snapbackend.timer file Type=simple WorkingDirectory=~ ProtectHome=true NoNewPrivileges=true ExecStart=/usr/bin/snapbackend ExecStop=/usr/bin/snapstop --timeout 300 $MAINPID User=snapwebsites Group=snapwebsites # No auto-restart, we use the timer to start once in a while # We also want to make systemd think that exit(1) is fine SuccessExitStatus=1 Nice=5 LimitNPROC=1000 # For developers and administrators to get console output #StandardOutput=tty #StandardError=tty #TTYPath=/dev/console # Enter a size to get a core dump in case of a crash #LimitCORE=10G  [Install] WantedBy=multi-user.target  # vim: syntax=dosini E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable) "I let him do it" and "I allow him to do it", why exactly does one require 'to'? Is there a way to detect the collapse in wave function of entangled particles instantaneously? Is there a machine or instrument which will notify us instantly when the wave-function has collapsed if we have access to only one of the entangled particles? Note that the wave function collapse will be triggered by observing the particle which we do not have. Or in other words is there a way to identify (instantaneously) if the entangled pair of a particle has been observed and the wave function has collapsed. SELinux preventing Apache from writing to a file SELinux is preventing the apache user from writing to a log file which it owns. When I do setenforce 0 it works. Otherwise it shows this error  IOError: [Errno 13] Permission denied: '/var/www/webapp/k/site/k.log'   The security context of the file:  $ ll -Z k.log  -rw-r--r--. apache apache system_u:object_r:httpd_sys_content_t:s0 k.log   The file was created when the SELinux mode was set to permissive.  How to set the security context so the apache user can write in that directory? I did set that directory security context using chcon but I can't find a suitable file type.  From audit.log:  type=AVC msg=audit(1409945481.163:1561): avc:  denied  { append } for  pid=16862 comm="httpd" name="k.log" dev="dm-1" ino=201614333 scontext=system_u:system_r:httpd_t:s0 tcontext=system_u:object_r:httpd_sys_content_t:s0 tclass=file type=SYSCALL msg=audit(1409945481.163:1561): arch=c000003e syscall=2 success=no exit=-13 a0=7fa8080847a0 a1=441 a2=1b6 a3=3 items=0 ppid=15256 pid=16862 auid=4294967295 uid=48 gid=48 euid=48 suid=48 fsuid=48 egid=48 sgid=48 fsgid=48 tty=(none) ses=4294967295 comm="httpd" exe="/usr/sbin/httpd" subj=system_u:system_r:httpd_t:s0 key=(null) tcolorbox manual tip box, how to clone? I would like to clone this box used on the tcolorbox manual, but I couldn't find the configuration in the manual.    Can you help me? I need to configure my server to permit the http delete method.  currently, I get the following error when I attempt to make this call:    HTTP/1.1 405 Method Not Allowed Date: Mon, 11 Apr 2011 20:56:23 GMT Server: Apache/2.2.17 (Win32) PHP/5.3.5 Allow: GET,HEAD,POST,OPTIONS,TRACE Content-Length: 233 Keep-Alive: timeout=5, max=98 Connection: Keep-Alive Content-Type: text/html; charset=iso-8859-1     I have searched far and wide and have not been able to come up with an answer as to how I can configure the server to allow this request.  Someone please help as it has put a stop to my project. Referencing the reference? How can I search the bash history and rerun a command? Can I search history in bash and run the result?