text:
40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
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ID Date Author Type Category Subject
13512   Sun Jan 7 03:22:24 2018 KojiUpdatePonderSqueezeDisplacement requirements for short-term squeezing

Interesting. My understanding is that this is close to signal recycling, rather than resonant sideband extraction. Is that correct?

For signal recycling, we need to change the resonant condition of the carrier in the SRC. Thus the macroscopic SRC length needs to be changed from ~5.4m to 9.5m, 6.8m, or 4.1m.
In the case of 6.8m, SRC legnth= PRC length. This means that we can use the PRM (T=5%) as the new SRM.

Does this T(SRM)=5% change the squeezing level?

13513   Sun Jan 7 11:40:58 2018 KevinUpdatePonderSqueezeDisplacement requirements for short-term squeezing

Yes, this SRC detuning is very close to extreme signal recycling (0° in this convention), and the homodyne angle is close to the amplitude quadrature (90° in this convention).

For T(SRM) = 5% at the optimal angles (SRC detuning of -0.01° and homodyne angle of 89°), we can see 0.7 dBvac at 210 Hz.

13514   Sun Jan 7 17:27:13 2018 gautamUpdatePonderSqueezeDisplacement requirements for short-term squeezing

Maybe you've accounted for this already in the Optickle simulations - but in Finesse (software), the "tuning" corresponds to the microscopic (i.e. at the nm level) position of the optics, whereas the macroscopic lengths, which determine which fields are resonant inside the various cavities, are set separately. So it is possible to change the microscopic tuning of the SRC, which need not necessarily mean that the correct resonance conditions are satisfied. If you are using the Finesse model of the 40m I gave you as a basis for your Optickle model, then the macroscopic length of the SRC in that was ~5.38m. In this configuration, the f2 (i.e. 55MHz sideband) field is resonant inside the SRC while the f1 and carrier fields are not.

If we decide to change the macroscopic length of the SRC, there may also be a small change to the requirements on the RoCs of the RC folding mirrors. Actually, come to think of it, the difference in macroscopic cavity lengths explains the slight differences in mode-matching efficiencies I was seeing between the arms and RCs I was seeing before.

 Quote: Yes, this SRC detuning is very close to extreme signal recycling (0° in this convention), and the homodyne angle is close to the amplitude quadrature (90° in this convention). For T(SRM) = 5% at the optimal angles (SRC detuning of -0.01° and homodyne angle of 89°), we can see 0.7 dBvac at 210 Hz.

13515   Sun Jan 7 20:11:54 2018 KojiUpdatePonderSqueezeDisplacement requirements for short-term squeezing

In fact, that is my point. If we use signal recycling instead of resonant sideband extraction, the "tuning" of the SRC is opposite to the current setup. We need to change the macro length of the SRC to make 55MHz resonant with this tuning. And if we make the SRC macro length together with the PRC macro length for this reason, we need to thing again about the mode matching. Fortunately, we have the spare PRM (T=5%) which matches with this curvature. This was the motivation of my question. We may also choose to keep the current SRM because of its higher T and may want to evaluate the effect of expected mode mismatch.

256   Wed Jan 23 12:31:36 2008 AndreySummarySUSDissapointing Results of XARM optimization (PDF-file)

I attach a PDF-file which summarizes briefly the results of measurements/calculations of Q-factors for ITMX mass as a function of suspension damping gain,

and this file contains the results of measurements of RMS peaks on the values of suspension gains of ITMX and ETMX (see ELOG entries from December 2007, specifically #202, #199, #194)),
but now those dependences are plotted in Q-ITMX and Q_ETMX axes.

Unfortunately, there are no clear narrow areas of minimum in those dependences (that explains the sad title of this ELOG entry).

The attached pdf-file can be shown as a short presentation for a wall during our Wednesday meeting.
8559   Thu May 9 15:07:51 2013 JenneUpdateGeneralDistances from CAD drawing

Since I keep asking Manasa to "measure" distances off of the CAD drawing for me, I thought I might just write them all down, and quit asking.

So, these are only valid until our next vent, but they're what we have right now.  All distances are in meters, angles in degrees.

4696   Wed May 11 23:02:52 2011 valeraUpdateASSDither angular stabilizitaion system update

This is what was done in past two days:

- The ETMY and ITMY pitch and yaw dofs are modulated at 40, 44, 42, 46 Hz respectively (oscillator A=30). The c1ass lockin numbers are 12, 14, 27, 29.

- The NAS55I signal is demodulated at the above frequencies. The demodulated I/Q signal phase is set to shift all signal into I-phase. The lockin inputs are bandpassed around respective frequency f with butter("Bandpass",2,f-0.5,f+0.5). The demod signals are then additionally low passed with butter ("Lowpass",4,0.5) so the servo ugf has to be below 0.5 Hz. The servo filter is p:z 0.0001:0.1.

- The ETMY demodulated signal is fed back to ITMY and visa versa.

- With the above 2x2 servo running we moved the input beam PZTs by hand to follow the cavity.

- At the end we offloaded the servo control signals to the SUS biases again by hand.

- The beam spot centering was estimated by unbalancing the ETMY/ITMY pitch/yaw coil combinations intentionally by 5%, which produces 1.3 mm shift of the node, and comparing the response to the residual signals.

- The dof set up currently is: ETMY pitch lockin 12 -> dof2, ITMY pitch lockin 14 -> dof4, ETMY yaw lockin 27 -> dof7, ITMY yaw lockin 29 -> dof9

- The next step is to demodulate the TRY(X) and servo the input beam PZTs

5165   Wed Aug 10 02:40:40 2011 JennyUpdatePSLDither freq for PZT chosen: 2.418 MHz

I've finished using the network analyzer to characterize find a dither frequency for driving the PZT to use in my PDH locking. I found a region in which the amplitude response of the PZT is low: The dip is centered at 2.418 MHz. Changing the NPRO laser temperature by 100mK has no significant effect on the transfer function in that region. I will post plots tomorrow.

I'm finished with the network analyzer. It is unplugged, and the cart is still near the PSL table. (I'll roll it back tomorrow when it won't disturb interferometer locking).

I closed the shutter on the NPRO at the end of the night.

Tomorrow I plan to put together the fast locking setup. I'll drive the PZT at 2.418 MHz. More details to come tomorrow.

10581   Wed Oct 8 03:20:46 2014 JenneUpdateLSCDo we need AO for acquisition?

As part of trying to determine whether we require the AO path for lock acquisition, or if we can survive on just digital loops, I looked at the noise suppression that we can get with a digital loop.

I took a spectrum of POX, and calibrated it using a line driving ETMX to match the ALSX_FINE_PHASE_OUT_HZ channel, and then I converted green Hz to meters.

I then undid the LSC loop that was engaged at the time (XARM FMs 1,2,3,4,5,8 and the pendulum plant), to infer the free running arm motion.

I also applied the ALS filters (CARM FMs 1,2,3,5,6) and the pendulum plant to the free running noise to infer what we expect we could do with the current digital CARM filters assuming we were not sensor noise limited.

In the figure, we see that the free running arm displacement is inferred to be about 0.4 micrometers RMS.  The in-loop POX signal is 0.4 picometers RMS, which (although it's in-loop, so we're not really that quiet) is already better than 1/10th the coupled cavity linewidth.  Also, the CARM filters that we use for the ALS lock, and also the sqrtInvTrans lock are able to get us down to about 1 pm RMS, although that is not including sensor noise issues.

For reference, here are the open loop gains for the LSC filters+pendulum and ALS filters+pendulum that we're currently using.  The overall gain of these loops have been set so the UGF is 150Hz.

It seems to me that as long as our sensors are good enough, we should be able to keep the arm motion down to less than 1/10th or 1/20th the coupled cavity linewidth with only the digital system.  So, we should think about working on that rather than focusing on engaging the AO path for a while.

Attachment 3: CARMnoise_7Oct2014.zip
1272   Wed Feb 4 19:22:57 2009 YoichiUpdateGeneralDo we need off-axis parabolic mirrors ?
I also estimated the mode matching degradation caused by the astigmatism.
Since the incident angles to the mode matching mirrors are not 0, the effective focal lengths in the incident plane and the perpendicular plane are different.
This effect leads to astigmatism of the beam.
When there is astigmatism, the maximum achievable mode matching rate becomes less than 100%.
According to my calculation, the mode matching cannot be better than 94% for the input beam.
For the output mode matching, we can theoretically achieve more than 99% even with the astigmatism.
The difference comes from the fact that the OMMT is longer, thus the incident angle is smaller.

If we don't like this 94%, we have to use off-axis parabolic mirrors, or modify the IMMT to a longer one.
I prefer to make it longer. Just 5" elongation will increase the mode matching rate to 99.4%.
We have a room for this 5" elongation.

Again, the details of the calculation are added to the Mathematica notebook below.

 Quote: I did mode matching calculations for the new optical layout. For the input mode matching, we have to change the focal length of the second mirror from 687mm to 315mm and the distance between the two MMT mirrors from 137mm to 149.2mm. For the mode matching to the OMC, we only have to change the distance between the OMMT mirrors from 384mm to 391mm. No need to change the mirrors. Details of the calculations can be found in http://lhocds.ligo-wa.caltech.edu:8000/40m/Upgrade_08/Optical_Layout?action=AttachFile&do=get&target=NewRecyclingCavities.zip (Mathematica notebook) or if you prefer PDF, here http://lhocds.ligo-wa.caltech.edu:8000/40m/Upgrade_08/Optical_Layout?action=AttachFile&do=get&target=NewRecCav.pdf
1274   Thu Feb 5 10:42:33 2009 YoichiUpdateGeneralDo we need off-axis parabolic mirrors ? No way !
I made a mistake in estimating the astigmatism problem.
If we use the current MMT1 as it is, this one is already an off-axis parabolic (OAP) mirror.
In this case, the astigmatism of this mirror is very small (if we use it with the correct angle). I did not include this effect in the previous calculation.
It turned out that the maximum achievable mode matching becomes far smaller (only 77%) if we use the OAP for MMT1 and a spherical mirror for MMT2.
This is not acceptable.
The reason behind this is that when we use spherical mirrors for both MMT mirrors, the astigmatism caused by the MMT1 is somewhat canceled by the astigmatism of MMT2. We don't get this cancellation if we mix OAP and spherical mirrors.

We should either (1) change MMT1 to a spherical mirror and keep the length of the input MMT as it is, or (2) change MMT1 to a spherical mirror and elongate the length of the input MMT.
In the case of (1) the maximum achievable mode matching is 94%. The focal length of MMT2 should be 315.6mm.
If we do (2), the mode matching rate can be as high as 99.8%. The focal lengths are MMT1 = -301.3mm, MMT2=558mm. The distance between the mirrors is 262mm.
We have enough space to do this elongation. But we have to mechanically modify the MMT mount.
I prefer (2).

As usual, the document on the Wiki was updated to include the above calculations.
11113   Sat Mar 7 18:53:48 2015 JenneUpdateLSCDoF selector matrices replaced with filter banks

This is work that I did yesterday but didn't have time to elog.  Since it seems non-trivial to give ourselves ramping matrices, but we only really needed the ramping in the DoF selector matrices, I've replaced the separate _A and _B parts with full filterbanks.  Recall from elog 10910 that I had given each degree of freedom's _A and _B input options an offset, an epics monitor and a test point.  Now those are removed, and handled inside of the filter banks.  The outputs of the filter banks sum together.

This required some screen modifications, but everything should work the same way that it did before this change.  I've also changed the DAQ channels from the _A_ERR and _B_ERRs that I had hand-created to now be the _A_OUT and _B_OUT test points from the filter banks (acquired at 2048Hz).

I have not yet modified the burt snapshots for the ifo configure screen.  The arms will work the same as always, since they didn't have any selector matrix stuff ever, but the rest still need tweaking.

Attachment 1: DoFselectors_7March2015.png
16249   Fri Jul 16 16:26:50 2021 gautamUpdateComputersDocker installed on nodus

I wanted to try hosting some docker images on a "private" server, so I installed Docker on nodus following the instructions here. The install seems to have succeeded, and as far as I can tell, none of the functionality of nodus has been disturbed (I can ssh in, access shared drive, elog seems to work fine etc). But if you find a problem, maybe this action is responsible. Note that nodus is running Scientific Linux 7.3 (Nitrogen).

13858   Thu May 17 13:51:35 2018 Jon RichardsonConfigurationElectronicsDocumentation & Schematics for AUX-PSL PLL

[Jon, Gautam]

Attached is supporting documentation for the AUX-PSL PLL electronics installed in the lower PSL shelf, as referenced in #13845.

Some initial loop measurements by Gautam and Koji (#13848) compare the performance of the LB1005 vs. an SR560 as the controller, and find the LB1005 to be advantageous (a higher UGF and phase margin). I have some additional measurements which I'll post separately.

## Loop Design

Pickoffs of the AUX and PSL beams are routed onto a broadband-sensitive New Focus 1811 PD. The AUX laser temperature is tuned to place the optical beat note of the two fields near 50 MHz. The RF beat note is sensed by the AC-coupled PD channel, amplified, and mixed-down with a 50 MHz RF source to obtain a DC error signal. The down-converted term is isolated via a 1.9-MHz low-pass filter in parallel with a 50 Ohm resistor and fed into a Newport LB1005 proportional-integral (PI) servo controller. Controller settings are documented in the below schematic. The resulting control signal is fed back into the fast PZT actuator input of the AUX laser.

## Hardware Photos

13876   Tue May 22 10:14:39 2018 Jon RichardsonConfigurationElectronicsDocumentation & Schematics for AUX-PSL PLL

Quote:

[Jon, Gautam]

Attached is supporting documentation for the AUX-PSL PLL electronics installed in the lower PSL shelf, as referenced in #13845.

Some initial loop measurements by Gautam and Koji (#13848) compare the performance of the LB1005 vs. an SR560 as the controller, and find the LB1005 to be advantageous (a higher UGF and phase margin). I have some additional measurements which I'll post separately.

## Loop Design

Pickoffs of the AUX and PSL beams are routed onto a broadband-sensitive New Focus 1811 PD. The AUX laser temperature is tuned to place the optical beat note of the two fields near 50 MHz. The RF beat note is sensed by the AC-coupled PD channel, amplified, and mixed-down with a 50 MHz RF source to obtain a DC error signal. The down-converted term is isolated via a 1.9-MHz low-pass filter in parallel with a 50 Ohm resistor and fed into a Newport LB1005 proportional-integral (PI) servo controller. Controller settings are documented in the below schematic. The resulting control signal is fed back into the fast PZT actuator input of the AUX laser.

## Hardware Photos

Attachment 1: Schematic_PLL.pdf
11823   Mon Nov 30 10:41:45 2015 yutaroUpdateLSCDoes a baffle in front of ETMY have effect on loss map measurement?

It might have, so I think I need to estimate shift of beam spot more preciely.

According to Steve's drawing, radius of the hole of the baffle is 19.8 mm.

Intensity distribution of fundamental mode in x axis direction is this (y is integrated out):

$I(x)=\frac{1}{\sqrt{2\pi(w/2)^2}}e^{-\frac{1}{2}\left(\frac{x}{w/2}\right)^2}$

With the radius of curvature of ETMY of 60 m and the arm length of 37.78 m, the beam width $w$ on ETMY is estimated to be 5.14 mm. From this expression of the intensity, $\int^\infty_{x=9.56\mathrm{mm}}\mathrm{d}xI(x)=100\,\mathrm{ppm},\,\int^\infty_{x=10.00\mathrm{mm}}\mathrm{d}xI(x)=50\,\mathrm{ppm}$, for example. If round trip loss is considered, these values are doubled.

Although maximum shift of beam spot from the ideal spot on ETMY is estimated to be sqrt(6.0^2+(1.7+1.7)^2)=6.9 mm, this value could have error of several tens of % because I am not sure to what exten the calibration is precise, which means that the maximum shift could be ~10 mm and seperation between the baffle and the beam could be ~10 mm.

Therefore, I need to check how much the beam spot shifts with another way, maybe with captured image of the CCD camera.

11828   Mon Nov 30 17:17:30 2015 yutaroUpdateLSCDoes a baffle in front of ETMY have effect on loss map measurement?

With captured images of ETMYF, I measured the shift of the beam spot on ETMY.

The conclusions are:

the baffle would have almost no effect on loss map measurement and

the calibration of beam spot shift is confirmed to be not so bad.

What I did:

I captured ETMYF images in the cases that (i)beam spot is centered on ETMY, beam spot is at the rightest and lowest point of my loss map measurement (corresponding to [0,0] component of the matrix shown in elog 11818), and beam spot is at the leftest and highest point of my loss map measurement ([4,4] component). Each captured image is attached.

Then using ImageJ, I measured the shift of the beam spot. I calibrated lengh in horizontal direction and vertical direction with the diameter of the mirror.

Results:

The amount of the beam shift was 7.2 mm and 8.0 mm for each case.

These values indicate that clapping loss due to the baffle is less than 10 ppm in a round trip.

Today's results support the previous calibration with oplev, which says the amount of the beam shift is 7.0 mm. Two values derived by different calibrations coincide within ~10 % though they are totally different methods. This also support the calibration of the oplev for ETMY (elog 11785) indirectly.

Attachment 1: element2-2.png
Attachment 2: element0-0.png
Attachment 3: element4-4.png
7321   Thu Aug 30 19:08:08 2012 JenneUpdateVACDogs on BS, ITM chambers checked

I tightened as many of the dog clamps on the bottom of the BS, ITMX and ITMY chambers as I could find.  I used a torque wrench at 45 ft-lbs.  Some of the bolts of the dogs were too long, and I couldn't find an extender thing to accommodate the bolt so I could reach the nut.  None of the bolts moved that I was able to reach.

Steve, we're not doing final final alignment today (we will do it tomorrow), so please go around and double-check my work by checking all of the dogs first thing in the morning.  Thanks.

14487   Wed Mar 20 12:31:30 2019 JonUpdateVACDoing vac controls work

I'm rebooting the IOLAN server to load new serial ports. The interlocks might trip when the pressure gauge readbacks cut out.

14489   Wed Mar 20 20:07:22 2019 JonUpdateVACDoing vac controls work

Work is completed and the vac system is back in its nominal state.

 Quote: I'm rebooting the IOLAN server to load new serial ports. The interlocks might trip when the pressure gauge readbacks cut out.

10029   Wed Jun 11 20:09:37 2014 ZachConfigurationWikiDokuWiki situation

I have been looking into the DokuWiki situation, with no great progress so far.

The problem is with authentication. When you try to access the wiki, you get: "User authentication is temporarily unavailable. If this situation persists, please inform your Wiki Admin."

As Evan points out, this goes away if you turn off ACL (Access Control Lists), but---as he also mentions---that just means that authentication is disabled, so the wiki becomes open. All signs point to this being a problem with the authentication mechanism. We use the 'authplain' plaintext method, where the usernames and hashed passwords are stored in a plaintext file.

Evan and I have both done plenty of testing with the config settings to see if the problem goes away. Even if you restore everything to default (but enable ACL using authplain and the existing user file), you still get an error.

I went as far as installing a brand new wiki on the web server, and, surprisingly, this also exhibits the problem. Interestingly, if I install an old enough version (2012-10-13), it works fine. After this revision, they transitioned their authentication methods from "backends" to "plugins", so this is a clue. As a side note, the other wikis on nodus (ATF and Cryo) are running earlier versions of DokuWiki, so they have no problems.

As it stood, our options were:

1. No wiki (not even read-only, since the issue prevents logging in and also prevents anonymous reading, somehow)
2. No ACL = open wiki. Also not good.

So, I created a temporary read-only version of the wiki using the aforementioned earlier DokuWiki release. It has a soft link to the real wiki data, but I deleted the user database so no one can log in and edit (I also disabled registration). It can be found at https://nodus.ligo.caltech.edu:30889/wiki_ro/.

I also created a backup of the real wiki as wiki_bak to avoid any potential disasters.

The simplest thing to do would be to just roll the wiki back to this working version, but that doesn't sound so smart. Clearly, it was working with the plugin structure before our snafu, and somehow our fixing everything else has broken it.

Quote:

 Quote: It looks like auth is broken on the AIC wiki (though working fine on ATF and Cryo). I did some poking around but can't see how anything we did could have broken it.

I went into local.php and changed $conf['useacl'] = 1; to$conf['useacl'] = 0; and it looks like the auth issue goes away (I've changed it back). This isn't a fix (we want to use access control), but it gives us a clue as to where the problem is.

10021   Tue Jun 10 19:11:27 2014 EvanConfigurationWikiDokuWikis are back up

As of this writing, the DokuWiki appears to be working.

As you and I suspected, it looks like this was a clusterwhoops with the permissions for the NFS mount. Let's recap what happened in the past 24 hours:

1. Yesterday, 8 PM: I restart the Apache server, thereby resurrecting the SVN (now conveniently located at /export/home/svn). The DokuWikis remain borked.
2. Yesterday, 7 to 11 PM: Zach, Rana, and Jenne perform deep magic to get the front-end machines up and running again. This should be irrelevant for this Apache/SVN/DokuWiki witchcraft.
3. Today, morning: the townsfolk happily resume their svn up and svn ci festival.
4. Today, ca. 3 PM: Zach runs stopapache.sh to bring down Apache, thinking he can bring it back up momentarily with startapache.sh. But NFS is a jealous and vengeful god, and Apache now complains that it doesn't have permission to write to its logfiles, and therefore can't start httpd.
5. Today, 5 PM: "How can this be?", Zach and I ask. Apache had no problem starting up yesterday night, and to our knowledge nobody futzed with chiara's NFS mount of /home/cds. This change remains mysterious to us.

Despite the aforementioned mystery, Zach and I pressed on and tried to diagnose the permissions issue. We found that even if you are logged into nodus or pianosa or rossa or whatever as the controls user, the NFS mount saw us as the user nobody (in the group nogroup). If we created a file on the NFS mount, it was owned by nobody/nogroup. If we tried to modify a file on the NFS mount that was owned by controls/controls or controls/staff, we got a "permission denied" error, even if we tried with superuser privileges.

It turns out this has to do with the vagaries of NFS (scroll down to gotcha #4). We have all_squash enabled in /etc/exports , which means that no matter your username or group on nodus, rossa, pianosa, or harpischordsa, NFS coerces your UID/GID to chiara's nobody/nogroup. Anyway, the fix was to go into chiara's /etc/exports and change this

/home/cds 192.168.113.0/255.255.255.0(rw,sync,no_subtree_check,all_squash,insecure)

to this

/home/cds 192.168.113.0/255.255.255.0(rw,sync,no_subtree_check,all_squash,anonuid=1001,anongid=1001,insecure)

where 1001/1001 are the UID/GID for chiara's controls/controls (as opposed to 65534/65534 for chiara's nobody/nogroup). That way, the NFS mount sees you as chiara's controls/controls.

In order to make chiara's NFS daemon aware of the new /etc/export settings, I ran sudo exportfs -r based on the answer to this StackOverflow question. As with all the best StackOverflow questions, the moderators closed it for being "off-topic".

[Edit, 2014-06-11, 11 AM: I've repeated the above anonuid/anongid change for the /home/cds/caltech/home/controls mount, so that nodus's /home/controls is writeable as well. I've also added a .screenrc for nodus in order to maintain sanity.]
10022   Wed Jun 11 05:16:14 2014 ZachConfigurationWikiDokuWikis are back up

It looks like auth is broken on the AIC wiki (though working fine on ATF and Cryo). I did some poking around but can't see how anything we did could have broken it.

 Quote: As of this writing, the DokuWiki appears to be working.

10024   Wed Jun 11 10:15:15 2014 EvanConfigurationWikiDokuWikis are back up

Quote:

It looks like auth is broken on the AIC wiki (though working fine on ATF and Cryo). I did some poking around but can't see how anything we did could have broken it.

 Quote: As of this writing, the DokuWiki appears to be working.

I went into local.php and changed $conf['useacl'] = 1; to$conf['useacl'] = 0; and it looks like the auth issue goes away (I've changed it back). This isn't a fix (we want to use access control), but it gives us a clue as to where the problem is.

10366   Mon Aug 11 23:50:38 2014 ranaConfigurationWikiDokuWikis are back up

Quote:

 Quote: It looks like auth is broken on the AIC wiki (though working fine on ATF and Cryo). I did some poking around but can't see how anything we did could have broken it.

I went into local.php and changed $conf['useacl'] = 1; to$conf['useacl'] = 0; and it looks like the auth issue goes away (I've changed it back). This isn't a fix (we want to use access control), but it gives us a clue as to where the problem is.

There was still some residual permissions issue. This is now bypassed and so the ACL is ON and all seems to be back the way it was. I've tested that I can login and edit the wiki.

Some useless knowledge follows here. Please ignore.

After some hours of reading unhelpful DokuWiki blogs, I just put the backup wiki into the local disk on NODUS and then made a soft link to point to that from /users/public_html/wiki/. So this implies that the new NFS setup on chiara is different enough that it doesn't allow read/write access to the apache user on the NODUS/Solaris machine.

12935   Mon Apr 10 15:22:46 2017 ranaConfigurationWikiDokuWikis are back up

AIC Wiki updated to latest stable version of DokuWiki: 2017-02-19b "Frusterick Manners" + CAPTCHA + Upgrade + Gallery PlugIns

10019   Tue Jun 10 11:54:36 2014 ZachConfigurationWikiDokuWikis are still down

Not sure if someone is already on this, but the nodus DokuWikis are still down (AIC, ATF, Cryo, etc.)

I get:

# DokuWiki Setup Error

The datadir ('pages') does not exist, isn't accessible or writable. You should check your config and permission settings. Or maybe you want to run the installer

4809   Mon Jun 13 15:33:55 2011 Jamie, JoeUpdateCDSDolphin fiber between 1Y3 and 1X4 appears to be dead

The fiber that connects the Dolphin card in the c1lsc machine (in the 1Y3 rack) to the Dolphin switch in the 1X4 rack appears to have died spontaneously this morning.  This was indicated by loss of Dolphin communication between c1lsc and c1sus.

We definitively tracked it down to the fiber by moving the c1lsc machine over to 1X4 and testing the connection with a short cable.  This worked fine.  Moving it back to using the fiber again failed.

Unfortunately, we have no replaced Dolphin fiber.  As a work around, we are stealing  a long computer->IO chassis cable from Downs and moving the c1lsc machine to 1X4.

This is will be a permanent reconfiguration.  The original plan was to have the c1lsc machine also live in 1X4.  The new setup will put the computer farther from the RF electronics, and more closely mimic the configuration at the sites, both of which are good things.

4815   Tue Jun 14 09:25:17 2011 JamieUpdateCDSDolphin fiber tested with c1sus, still bad

The bad Dolphin was still bad when tested with a connection between c1sus and the Dolphin swtich.

I'm headed over to Downs to see if we can resolve the issue with the PCIe extension fiber.

4045   Mon Dec 13 11:56:32 2010 josephb, alexUpdateCDSDolphin is working

## Problem:

The dolphin RFM was not sending data between c1lsc and c1sus.

## Solution:

Dig into the controller.c code located in /opt/rtcds/caltech/c1/core/advLigoRTS/src/fe/.  Find this bit of code on line 2173:

2173 #ifdef DOLPHIN_TEST 2174 #ifdef X1X14_CODE 2175         static const target_node = 8; //DIS_TARGET_NODE; 2176 #else 2177         static const target_node = 12; //DIS_TARGET_NODE; 2178 #endif 2179         status = init_dolphin(target_node);

Replace it with this bit of code:

2173 #ifdef DOLPHIN_TEST 2174 #ifdef C1X02_CODE 2175         static const target_node = 8; //DIS_TARGET_NODE; 2176 #else 2177         static const target_node = 4; //DIS_TARGET_NODE; 2178 #endif 2179         status = init_dolphin(target_node);

Basically this was hard coded for use at the site on their test stands.  When starting up, the dolphin adapter would look for a target node to talk to, that could not be itself.  So, all the dolphin adapters would normally try to talk to target_node 12, unless it was the X1X14 front end code, which happened to be the one with dolphin node id 12.  It would try to talk to node 8.

Unfortunately, in our setup, we only had nodes 4 and 8.  Thus, both our codes would try to talk to a nonexistent node 12.  This new code has everyone talk to node 4, except the c1x02 process which talks to node 8 (since it is node 4 and can't talk to itself).

I'm told this stuff is going away in the next revision and shouldn't have this hard coded stuff.

## Different Dolphin Problem and Fix:

Apparently, the only models which should have pciRfm=1 are the IOP models which have a dolphin connection.  Front end models that are not IOP models (like c1lsc and c1rfm) should not have this flag set.  Otherwise they include the dolphin drivers and causes them and the IOP to refuse to unload when using rmmod.

So pciRfm=1 only in IOP models using Dolphin, everyone else should not have it or should have pciRfm=-1.

Current CDS status:

 MC damp dataviewer diaggui AWG c1ioo c1sus c1iscex RFM Dolphin RFM Sim.Plant Frame builder TDS
5841   Tue Nov 8 17:48:21 2011 MirkoUpdateCDSDolphin weirdness

Had since yesterday evening some trouble with getting a channel from rfm on c1sus to oaf on c1lsc via dolphin. Several restarts of c1lsc and c1sus didn't help. At some point this morning a restart of c1lsc helped. Everything ok again.
At the bad time the dolphin TF looked like this:

Should be flat at gain 1 and no phase change obviously.

184   Mon Dec 10 13:54:26 2007 robHowToComputer Scripts / ProgramsDon't blame the Matlab compiler

For human error. We should be careful to only run the compiler outside the mDV directory (thus placing the _mcr outside of the range the addpath command in the mdv_config files). Or maybe there's a smarter solution...
4985   Mon Jul 18 21:06:32 2011 PSL Table GuardianOmnistructurePSLDon't leave the PSL table open, unattended!!!!!!!!!!!!11111

I found the PSL table left open, and unattended again.

As far as I know, Jamie and Jenne (working on the LSC rack, so no lasers / optics work involved) have been the only ones in the IFO room for several hours now.

I'm going to start taking laser keys, or finding other suitable punishments.  Like a day of lab cleanup chores or something.  Seriously, don't leave the PSL table open if you're not actively working on it.

602   Mon Jun 30 13:48:47 2008 John, RobConfigurationPSLDon't put the bin in front of the air conditioning unit
We spotted that the laser power was dropping.

The air conditioning unit was blocked by the blue bin/trash can/cestino causing the laser head temp to increase by 2 degrees.

Attachment 1: binremoval.png
603   Mon Jun 30 14:07:26 2008 RobConfigurationPSLDon't put the bin in front of the air conditioning unit

 Quote: We spotted that the laser power was dropping.

the offending configuration:
Attachment 1: DSC_0140.png
12376   Thu Aug 4 17:57:09 2016 KojiConfigurationGeneralDon't restart apache2 - nodus /etc/apache2/sites-available/* accidentally deleted

Late coming elog about the deletion of the apahce config files

Thu Aug 4 8:50ish 2016

I accidentally deleted four files in /etc/apache2/sites-available / on nodus. The deleted files were

elog   nodus  public_html  svn

I believe public_html is not used as it is not linked from /etc/apache2/sites-enabled

They are the web server config files and need to be reconfigured manually. We have no backup.

Currently all the web services are running as it was. However, once apache2 is restarted, we'll lose the services.

7278   Sun Aug 26 20:58:21 2012 JenneUpdateVACDon't vent!!!!

[Koji, Jenne]

Steve, do not vent tomorrow morning!  We are still not prepared, and will not finish the preparation tonight.  Hopefully we can finish the prep tomorrow, and then vent Tues.

Things we need to do before the vent:

MC spots centered [Jenne, tonight]

Use PZT2, BS to hit ~center of ETMs.

Realign arms, measure spot positions.

Make sure BS, ITMs are good - we want a good AS spot since we'll likely have to adjust some AS optics while we're inside

Insert attenuation optics, recover MC trans by rotating PBS cube to translate beam slightly

2961   Thu May 20 20:03:37 2010 JenneUpdateSAFETYDon't walk down the Y(new convention) arm tonight!

Please don't go down the Yarm (Old Xarm) for right now, or if you do, please be very careful.  Kiwamu and I are set up to take beam scan measurements down the walkway, and so there are some cables / carts / other stuff down there.  We are going to get dinner really quickly before beginning the measurements.

Right now, the PSL shutter is Closed, so there is no beam hazard outside of the chambers, just crowded space hazard.

2962   Fri May 21 00:21:24 2010 JenneUpdateSAFETYDon't walk down the Y(new convention) arm tonight!

All clear.

 Quote: Please don't go down the Yarm (Old Xarm) for right now, or if you do, please be very careful.  Kiwamu and I are set up to take beam scan measurements down the walkway, and so there are some cables / carts / other stuff down there.  We are going to get dinner really quickly before beginning the measurements.  Right now, the PSL shutter is Closed, so there is no beam hazard outside of the chambers, just crowded space hazard.

11837   Wed Dec 2 15:08:41 2015 ericqUpdateComputer Scripts / ProgramsDonatella sudo problem resolved

Somehow, the controls user account on donatella lost its membership to the sudoers group, which meant doing anything that needs root authentication was impossible.

I fixed this by booting up from a Linux install USB drive, mounting the HD, and running useradd controls sudo

14578   Thu Apr 25 18:14:42 2019 AnjaliUpdatePSLDoor broken

It is noticed that one of the doors (door # 2 ) of the PSL table is broken. Attachement #1 shows the image

Attachment 1: IMG_6069.JPG
16867   Fri May 20 12:42:23 2022 TegaUpdateVACDoor installation on the end stations

[JC, Tega, Chub]

Today we installed the 200 lbs doors on the end station chambers.

15932   Wed Mar 17 15:02:06 2021 gautamUpdatesafetyDoor to outside from control room was unlocked

I came into the lab a few mins ago and found the back door open. I closed it. Nothing obvious seems amiss.

Caltech security periodically checks if this door is locked but it's better if we do it too if we use this door for entry/exit.

15938   Thu Mar 18 12:35:29 2021 ranaUpdatesafetyDoor to outside from control room was unlocked

I think this is probably due to the safety tour yesterday. I beleive Jordan showed them around the office area and C&B. Not sure why they left through the control room.

 Quote: I came into the lab a few mins ago and found the back door open. I closed it. Nothing obvious seems amiss. Caltech security periodically checks if this door is locked but it's better if we do it too if we use this door for entry/exit.

7432   Mon Sep 24 16:59:31 2012 JenneUpdateVACDoors on, ready to pump

[The 40m Family]

The access connector and all heavy doors are back on.

Jamie put the regular viton EQ stops back on PRM, since he had to adjust the distance between the EQ stops and the PRM anyway.  Jamie also waved an IR card near the IPANG steering mirrors in ETMY, but it was not possible to take a good photo.  Jamie certifies that the beam is centered on both of those 2 optics.

I have centered IPPOS and IPANG QPDs.

All oplevs need a little realignment, especially ETMY, which had it's lens removed (Rana has a Wall of Shame photo of this, which is why it was removed by him).  Steve will look into this tomorrow, after he starts pumping.

I have turned off all PZT high voltage supplies for in-vac PZTs: The input PZTs, the output PZTs, and the OMC PZTs (which weren't on, but I confirmed they were off).

I have also prepared the 3 low-power items for high power: MC refl's path was changed back to regular BS, AS camera was moved to its nominal position, and IPPOS has its ND filters back.  MC refl and the AS camera will need to be realigned once we're actually at high power tomorrow afternoon.

Long vent, but good work everyone.

 Quote: All oplevs need a little realignment, especially ETMY, which had it's lens removed (Rana has a Wall of Shame photo of this, which is why it was removed by him).  Steve will look into this tomorrow, after he starts pumping.

The shame:

There is no situation in which it is OK to install a mount like this. Steve had installed this flaky and shaky mount to optimize the beam size on the OL QPD.

Everyone in the lab should know better. Putting in something like this is just like sabotage - it creates extra noise in our interferometer in a sneaky way and just makes locking harder. All mounts for anything useful (including QPDs) must have highly rigid mounts.

Use the example from the PSL relayout: use the 3/4" steel mounts and the wide aluminum bases from Newport. No more art projects using home made mounting crap, Steve.

7250   Wed Aug 22 16:50:09 2012 JenneUpdateLockingDouble integrator in ARM LSC servos

Last week, Rana changed the integrators in the arm LSC servo filters to be double integrators with complex poles.

Yesterday, I found that using the "timeout" feature of Foton (at filter ON/OFF request, waits for zero crossing, or T seconds, whichever comes first) is useful for turning on the integrators, but bad for turning them off.  When we're locked, the error signal is oscillating around zero, so there is often a zero crossing.  When we lose lock, we want to turn off the filter immediately.  But, as soon as lock is lost, the input signal gets large, and doesn't often cross zero, so the filter waits 8 seconds until actually turning off.  If the arm flashes any time during that 8 sec, we send a big kick to the optics.

An alternative option could be ramping the filter on.  However, since the double integrator has -180deg phase at low frequencies (until the poles at ~5Hz), the transition between no filter (0deg phase) and integrator on could be problematic.  I simulated this, and find that for the very beginning of the ramping process, we would have a problem.

The filter is defined as:  NoFilter * (1 - R) + Integrator * (R), so for R=0, the integrator is off, and for R=1, the integrator is fully on.  R can be any value [0,1].

The first figure is the time series (1 second, 16kHz), ramp goes from 0->1 or 1->0 in 1 second:

The second figure is bode plots for selected values of R:

As R gets smaller and smaller, the notch goes to lower frequency, and becomes higher Q.  So perhaps ramping is not a good answer here.

What if we go for single or triple integrator, to get rid of the (+1) + (-1) problem?

11516   Wed Aug 19 01:45:10 2015 IgnacioUpdateIOODoubly Improved SISO (T240-X) FF of MCL

Today I tried and doubly-improved SISO FF filter on MCL. This filter has a stronger rolloff than the previous SISO filters I have tried. The rolloff most definelty helped towards reducing the ammount of noise being injected into YARM. Below is the usual stuff:

Filter:

T240-X (SISO)

Training data + Predicted FIR and IIR subtraction:

Online subtraction results:

MCL
YARM
MCL TRANS

Subtraction performace:

3583   Fri Sep 17 12:11:42 2010 josephbUpdateCDSDowns update

In doing a re-inventory prior to the IOO chassis installation, I re-discovered we had a missing interface board that goes in an IO chassis.  This board connects the chassis to the computer and lets them talk to each other.  After going to Downs we remembered Alex had taken a possibly broken interface board back to downs for testing.

Apparently the results of that testing was it was broken.  This was about 2.5 months ago and unfortunately it hadn't been sent back for repairs or a replacement ordered.  Its my fault for not following up on that sooner.

I asked Rolf what the plan for the broken one was.  His response was  they were planning on repairing it, and that he'd have it sent back for repairs today.  My guess the turn around time for that is on the order of 3-4 weeks (based on conversations with Gary), however it could be longer.  This will affect when the last IO chassis (LSC) can be made fully functional.  I did however pickup the 100 foot fiber cable for going between the LSC chassis and the LSC computer (which will be located in 1X3).

As a general piece of information, according to Gary the latest part number for these cards is OSS-SHB-ELB-x4/x8-2.0 and they cost 936 dollars (latest quote).

5824   Sun Nov 6 16:58:25 2011 ZachUpdateSUSDr. SUS failed--NDS2 problems again

Dr. SUS failed while trying to get the sensor data. Specifically, it couldn't get ETMY data. This is odd, because in my tribulations last week I ended up having to add the ETMY_SENSOR channels manually into the NDS2 channel files. After doing this, I was able to get ETMY data just fine (though I admitted that we would have problems again as soon as we wanted to update the channel files). I even ran the diagAllSUS code in a sandbox and it pulled data---and generated input matrices---just fine.

The error persists even if I try to get the data manually:

>> d = NDS2_GetData({'C1:SUS-ETMY_SENSOR_UL'},t0,10,'mafalda.martian:31200');

Connecting.... authenticate done

Warning: daq_recv_next failed

??? Error using ==> NDS2_GetData

Fatal Error getting channel data.

I think Jamie is still waiting for J.Z.'s help with this, but it is probably pointless to keep trying to run this code before NDS2 is working again. Another option is to just use NDS, but I think certain people are opposed to this.

ELOG V3.1.3-


text:
# Quantify the asymmetry of a one dimensional vector

I have image data, which I can represent as one dimensional vectors. Each value represents the brightness of a pixel in a line.

eg:

(1, 12, 4, 3, 1, 4)

I want to describe the distribution of brightness levels in the vector with single digit summary statistics: how far and in which direction the "bright part" of the image is away from the centre, and how broad the "bright part" is. Together they will describe a region of interest in the underlying image. In general I expect the bright area to be a unimodal peak, or if not, then a diffuse plateau, so the description of these properties doesn't need to be complex.

The vectors will be of an unknown length, so a normalised range of values would be ideal as an output.

I thought of skewness and kurtosis as they seem conceptually similar (at least graphically), but as Nick said below these have no spatial or ordered component. I thought I could use the vector indices to transform my data, and apply a "spatial skewness/kurtosis", but I can't work out how to do this.

I would prefer to use something that seems familiar to my audience, which will largely be medical professionals who have varying amounts of statistical knowledge. In this way, a "spatial skewness" would be a recognisable idea, where another method of describing things may not be.

I am not wedded to this method however (it may not even be possible). If there is a good approach that won't be too computationally intensive I am open to that too.

edit:

OK, so having thought about this a bit more I think I can try to explain more clearly my purpose. I am trying to efficiently describe the spatial distribution of values in a vector to supply to a machine learning model. The actual vectors in my data are hundreds of numbers long, and contain a lot of noise, so I am essentially trying to smooth and perform dimensionality reduction by describing the overall distribution succinctly.

The weighted mean position is great because it describes where the "mass" of values is in the vector, but it obviously doesn't describe the distribution in other ways. In particular, the shape of the distribution is missed, whether the distribution is peaked or plateau, unimodal or multimodal etc.

In an effort to not overthink things, the solution that comes to me is to also do some binned averages ie average of first 25% of vector, average of second 25% etc.

The only issue I have with this is the inelegance of it. I am using multiple values to describe one "idea", and if I want an even more fine-grained description, I add more values by making the bins smaller. This is why I had gravitated to the concept of kurtosis initially, because it describes something about shape in a single number. I realise it is not applicable at all, but I still wonder if there is a more elegant solution to describing the shape of my distributions.

I understand this is what Nick has hinted at below, with "other weighted moment measures". I am again just running against my lack of mathematical intuition. I am not sure how to make a weighted second, third or fourth moment, let alone have a grasp of which one might be useful here.

• Skewness is defined for a set of magnitudes. If you've seen formulas involving frequencies, they are recipes for binned data. Any decent statistical program will give you skewness directly for a vector like yours. Watch out that there are slightly different formulas. Kurtosis doesn't measure asymmetry. But -- most important -- what you want doesn't seem to be either. Averaging the positions on your line with weights given by brightness may be more like it. Thus 255, 0, 0, 0, ... would return an average of position 1 and 0, 0, 0, ..., 255 would return an average of the last position. – Nick Cox Jan 15 '16 at 12:44
• (I am guessing very wildly at measurements 0 to 255 but the principle is the same for non-negative values.) – Nick Cox Jan 15 '16 at 12:45
• Hmm... This is why I felt confused, I thought skew should work out of the gate. It was just giving me bad numbers when I tested it. I'll retry. Thanks – Luke_radio Jan 15 '16 at 12:52
• Skewness won't take any account whatsoever of the positions of the measurements. For example, the skewness is the same if you reverse the vector from last to first or shuffle the elements randomly. So, it's not what you want if I understand you correctly. – Nick Cox Jan 15 '16 at 12:54
• Concerning asymmetry, are you perhaps asking the same thing addressed in the thread at stats.stackexchange.com/questions/145296? Please note that the question of "main peak ... high and sharp" is a completely different issue. It is rare that anyone would even care about that; often sharpness of the peak is used as a visual proxy for how long the tails of the distribution are. Could you perhaps tell us in more general, non-statistical terms what features of each image you are trying to describe or learn about? – whuber Jan 15 '16 at 15:41

I am still sketchy on what you want but guess that you are over-thinking this, as the similarity of skewness calculation to your problem is less than it seems. Kurtosis seems even less relevant.

Given brightness along a line -- in practice measured as positive integers, but my argument applies to any non-negative measure -- you (seem to) want to quantify how far values are high on the left compared with the right, and vice versa.

Keeping for simplicity to your positions 1 to 6, and using your toy examples (small detail: one is for 7 values, not 6) it seems to me that weighted mean position will do what you want. The weights are brightness measures; what is being averaged is the position. For example, (20, 0, 0, 0, 0, 0) at positions 1 to 6 will return a mean of 1 and flipping it round will return a mean of 6. A constant brightness will return a mean of 3.5.

Rather than working through trivial calculations, let's just show how the weighted mean positions work out. Here the bars show brightness; the small gap between bars is just cosmetic and could just be removed if anyone insists fairly that pixels are to be considered as touching. The spikes show the means.

For a more general situation, scaling by

2 $\times$ (weighted mean position $-$ mean position) / (length of line $-$ 1)

will produce a measure from $-$1 (high values at extreme left) through 0 to 1 (high values at extreme right). Simple variations on such a theme are naturally possible. In particular, its absolute value quantifies departure from the centre regardless of left or right.

Similarly, other weighted moment measures may be useful.

EDIT: Factor of 2 introduced into displayed equation.

• I suspect you are right that I am over-thinking it. I will think about the weighted mean, when you suggested it earlier I tried it briefly and thought there might be some problems with it, but I will work through it properly with my data and see if it does what I need. Cheers. – Luke_radio Jan 19 '16 at 23:42
• I have gone through this and it definitely does part of what I am looking for, so I am happy to tick your answer. I have added a bit more to my original question about what is left of my problem if you or anyone else felt inclined to assist further. Thanks. – Luke_radio Jan 23 '16 at 11:16
• I'll try to come back to this soon. As an interim comment, I would suggest the weighted SD of position and weighted quantiles (e.g. deciles, quartiles, median) of position as the easiest elaborations or alternatives to my first suggestion. Chopping each line into 4 parts sounds more arbitrary. – Nick Cox Jan 23 '16 at 18:12

I know I did not ask this question well, but I have found what I think is the start of an answer. I am posting it for feedback/review, in case it is useful to anyone else in the future. It does piggyback on skewness and kurtosis.

I would appreciate any attempts to poke holes in it. It works for some toy problems, but has some weaknesses.

I find the midpoint of the line (the middle value) and index each position relative to that. So in my above example:

x =  (1, 12, 4, 3, 1, 4)

the original index, length and midpoint are:

y =  (1, 2, 3, 4, 5, 6)
length_y = 6
midpoint_y = length_y/2 + 0.5

then:

(y - midpoint_y)

gives me a relative-to-midpoint range for my index, in this case:

new_index = (-2.5, -1.5, -0.5, 0.5, 1.5, 2.5)

I can then transform each value in the original vector with its new index position (image data is only positive integers, so this should work fine):

x * new_index = (-2.5, -18.0, -2.0, 6.5, 1.5, 10.0)

I take this result and run the normal Pearson skewness (third moment/second moment$^{3/2}$) which returns:

-0.7898

If this was normal skewness, the distribution would have a long tail to the left. That doesn't seem reassuring, because spatially the tail is to the right. But more testing shows:

input vector = (2,4,2,1,4,18,14)
output = 0.8561

input vector = (1,12,14,13,1,4)
output = -0.5278

input vector = (12,10,5,5,10,12)
output = 0

Which looks sign flipped. So if I just multiply by -1, it looks like a good start on a "spatialised skewness" measurement.

So far, looks OK. The bigger the values and the further from the middle, the bigger the output. But it falls apart again because the spatial order isn't maintained in the skewness calculation. If there are lower values outside the "peak", the output value will decrease as the end value goes up!

for example:

input vector = (1, 12, 4, 3, 1, 4)
output = 0.7898

input vector = (2, 12, 4, 3, 1, 4)
output = 0.6267

input vector = (3, 12, 4, 3, 1, 4)
output = 0.4767

which is not the output I am looking for. The overall "weight" of the numbers should be skewing the output value further, but because the further left value (when multiplied by its index position) is smaller than the 2nd from the left value, it actually pulls the skew back to the centre.

The other problem is that the output value increases as the vector length increases, meaning I have no standardised values for comparison if vector lengths are different (they are!). I'm not sure what to do about that either.

I could be barking up the wrong tree entirely here. Hopefully if nothing else, this better explains what I am trying to do.

P.S. I am ignoring kurtosis for the moment. The current system applies to that too, but likewise fails in funny ways.

format: plain


text:
# Lagrangia is it invariant

1. Jan 19, 2014

### shounakbhatta

Hello,

Is Lagrangian invariant?

I am in a conversation, where one is saying that:

"Shifting the coordinate system changes the value of the potential energy with respect to the same reference level, that's why the Lagrangian changes"

While the other:

"Shifting the coordinate system, doesnt shift the zero reference level of potential for the system..
The reference level of potential is fixed for the system once it is decided upon."

So, is Lagrangian variant or invariant?

Thanks.

2. Jan 20, 2014

### maajdl

The Lagrangian has the symmetries of the system that it represents.

3. Jan 20, 2014

### shounakbhatta

In the case can we consider that is it variant or invariant?

Also, is it a scalar or a vector product?

4. Jan 20, 2014

### BruceW

it depends what the Lagrangian is. some Lagrangian equations will be variant and some will be invariant.

edit: the definition of invariant, is that if you make some change, then the Lagrangian will still look the same. So if you shift the coordinate system but the Lagrangian still looks the same, this means the Lagrangian is invariant with respect to a coordinate shift.

5. Jan 20, 2014

### shounakbhatta

If it is a variant in a equation then it cannot be scalar?

Is it that it becomes scalar as well as a vector?


text:
## What is probability?

• 4
Well, it's an awkward question, but, what in fact is probability? I mean, we assume that if an event has probability of 99,9% of happening, it means that if we simulate the conditions, each 1000 times the event would occur next to 999 times. But that's not a fact, since nothing really prohibits the complement of the event, with probability of 0,1% of keep continuously occurring through time, while the first event, with almost 100% of probability never happens.
Anyone could argue that this is not likely, or if it happens, if you repeat the experiment, it would probably not happen again. But those two arguments just use again the definition of probability without explaining it.
I could add that saying "Means that something is more or less likely" just change the word probable for likely.
I know that in real life events like the first described are not usual, but mathematically it's not impossible and it's just a scenario to the main question: What, conceptually, is probability? What is something being likely to happen?
I don't know if I expressed my question in a proper way, but we can discuss to make it clear if that's a point.
• 595

I think this question goes far than mathematical criteria. Probably we should focus in terms of luck.
Probabilities and chances depend about of how we want to assume if the goals/recognition we get depends somehow of “luck”
How many luck do I have in terms of passing the next exam? It depends in how increases our probabilities while we study more or less.
Conceptually, when we check the definition of probability in Oxford dictionary, it redirects you synonym that is called “likelihood” which says the chance of something happening; how likely something is to happen

It is interesting how literally says “chance”. This is why I guess the significance of probability depends a lot of “opportunities to do something”
• 4.8k
I could add that saying "Means that something is more or less likely" just change the word probable for likely.

All definitions are just changing one word that you don't know the meaning of for another that you do.

A probability that an event P will have a result R is X if, when P takes place N times, on average, R will result X * N times.
• 1.6k
Well, it's an awkward question, but, what in fact is probability?

It's a very legitimate question that has received plenty of attention from mathematicians (who invented the concept) and philosophers. Have a read: Interpretations of Probability
• 4
• 156
What, conceptually, is probability? What is something being likely to happen?

Statistical probability is (for all necessary objectives) a formalism, that underlies the intuitive notion of a 'likelihood' (as you've discussed).

I mean, we assume that if an event has probability of 99,9% of happening, it means that if we simulate the conditions, each 1000 times the event would occur next to 999 times. But that's not a fact, since nothing really prohibits the complement of the event, with probability of 0,1% of keep continuously occurring through time, while the first event, with almost 100% of probability never happens.

You've identified precisely why probabilities aren't exhaustive, in and of themselves - they don't preclude the complement of an event (or an event itself), irrespective of how unlikely (or likely) they exert themselves to be. Unless they're of an unequivocal certainty (0 or 1), they shouldn't be conflated with assertions of certainty - or its lack, thereof.

Probabilistic interpretations, nevertheless, often 'suggest' that an assertion - of likelihood - is of practical utility in a decision-making paradigm (or likewise, not).

In this particular analogy, one can formalize a probabilistic construct with a binomial distribution for the complement of a 'significantly probable' event - denoted by $B (1000, 0.001)$, in the characteristic form $B (N, p)$.

You might be acquainted, in some measure, with the following sequence of reasoning:

For any elective integer $K$, wherein $0 \leq K \leq 1000$, there exists a discernible probability that conducting 1000 iterations of an arrangement, will result in $K$ complements to an event.

Specifically, if $P (N, K, p)$ illustrates the likelihood of an outcome (under the purview of a sample space) with an affiliated probability $p$, emerging on $k$ occasions after exactly $N$ iterations - then a canonical inference entails:

$P (N, K, p) = \binom{N}{K} [p^{K}] [(1-p)^{{N-K}}]$

Repurposing this expression, by virtue of the aforementioned system's $n$ and $p$ constituents suggests:

$P (1000, K, 0.001)= \binom{1000}{K} [0.001^{K}] [0.999^{{1000-K}}]$

0.001 is an imperceptible ratio, but is nonetheless fathomable. For instance, when $a = 1000$ (an indescribably anomalous iteration set), $P (N, K, p) = 0.001^{1000}.$

This might be misleading, unfortunately - since singular or incremental probabilities are nearly always inconsequent.

One may consolidate (or assimilate) the singular probabilities of $K$ variable iterations, across the discrete interval $a \rightarrow b,$ in order to accumulate superior likelihoods. Naturally, this engenders a binomial cumulative distribution - for a complement to manifest anywhere between $a \leq K \leq b$, when a probabilistic event is undertaken with $N$ distinctive iterations.

What if one were to redefine (for argumentation) $N=1000,$ to $N=1000^{100}$ - whilst $a$ and $b$ were unchanged? We'd witness an inconceivably larger number of operations (or iterations), therefore bolstering the likelihood of spectating a particular outcome across a specific interval of frequencies.

In answering your question, probability - schematically - is merely a suggestive trajectory to how 'likely' an event should be construed as being. It's neither infallible, nor overly authoritative; it exists solely to guide.
• 4
Changing the words of a phrase by it's synonyms may help to understand some concepts by approaching the speech to something more familiar to the audience, but that's not a definition at all, it's not adding information.

"A probability that an event P will have a result R is X if, when P takes place N times, on average, R will result X * N times" -> I don't know yet how to quote as fancy as you do, and would appreciate the help to learn!

But, life goes on: I don't know if you are saying that this is a definition, but assuming you are, it is just describing how to get to the magnitude of the probability of an event. It is like saying that price is the law of supply and demand.
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Thank you, I really appreciated your explanation. This last part 'suggestive trajectory to how 'likely...', it's tricky to me, 'cause in my language the translation of likely and probable is the same, so looks like not going further from the initial point. But adding this information 'suggestive trajectory...' was enough to make think a little more.
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Thank you, I really appreciated your explanation. This last part 'suggestive trajectory to how 'likely...', it's tricky to me, 'cause in my language the translation of likely and probable is the same, so looks like not going further from the initial point. But adding this information 'suggestive trajectory...' was enough to make think a little more.

Quite truthfully, after engaging in this discussion, I'm not sure I understand probabilities as well as I thought I did.

You're right - I haven't been able to further your initial argument. Personally, I don't know if one can.

Mathematically, one can always devise one sophistication upon another.

Philosophically, however - determining what exactly likelihood entails, is a genuine conundrum.
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I have a rather layman understanding of 20th century physics, but this is evidently important question for QM. Without philosophically meaningful description of probability, using propensities as subject matter is vague.

I have some conclusions of my own that I will propose.

Basically, the mathematical definition, specifically the one about expectation, aims to compute utilities over collections of instances. It deals with other quantities as well, but those are usually advisory, and the central notion is the mean. There is no need for non-determinism or incomplete knowledge to use probability. You could compute means over populations in some country or over the last month's orders of a company, etc. As long as you know the frequencies in advance, by direct measurement, the results are infallible.

When you don't have the ability to measure probability directly, because you either lack knowledge or there is genuine non-determinism, there are multiple avenues to take.

One involves past experience and statistics. The root of confidence in any kind of inductive practice is our history, both personal and collective, both recorded and genetic. Every time we achieve success, we experience confirmation bias and further ourselves, socially and as genetic viability. This develops and reinforces our habit, culture, and heritage. Like anything in nature, statistical induction will either work or will not, but from what we have seen, in most cases, had we decided not to trust the probabilistic consistency of nature, or the uniformity of nature, we would have had retroactively lost from the outcome. So, our actions are justified only in retrospect. Statistical thinking is in our natural programming, social convention, and habit.

There are other ways to extract distributions. You can seek trivial probability events (certain) and derive the rest of the distributions for related properties by first principles. For a dice roll, considering that the dice is symmetric, and since the roll is a certain event, each side would be presumed to roll with equal sixth chance. Similarly, if you toss three coins, you can derive the outcome chance of each configuration by independence. Starting with certain event and following symmetry and independence arguments, you can derive a lot of event chances.

The last method involves Bayesian thinking, where starting with particular distribution, either of the initial conditions in some stochastic process, or initial constraint on the present event, you reevaluate your probability constantly.

So much for now. Article on the Stanford E. P. that deals with statistics. https://plato.stanford.edu/entries/statistics
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A while back someone on the site told me the technical names for what I call ontological and epistemological probability but I don't seem to remember them at the moment so I'll just use my terms.

Say you flip a coin. If we believe in Newtonian physics for macroscopic events, the result of the coin flip is absolutely determined by the initial state and exact position of the coin, the flipping force, the air pressure, local gravity (accounting for our local elevation, say), the humidity of the air, and so forth. If we only knew all of those factors, we could predict the outcome of the coin toss with perfect accuracy.

The problem is, we can't know all those things with sufficient accuracy The coin toss is random not because it's inherently random, which it isn't. The coin toss is random because we can't possibly know the factors that go into its determinate outcome. That's what I call epistemological randomness. The event is random because we don't know, or we can't know, enough about the factors that cause it to be one way or the other.

Now a moment's thought reveals that pretty much everything that we regard as probabilistic is actually of this type. It's very difficult to think of anything at all that's truly random -- ontologically random -- which I define as random not because of our ignorance of the deterministic factors, but actually deep down random.

We are hard pressed to give even a single example of an ontologically random event. Most people will fall back on quantum events. The low-order bit of the femtosecond timestamp of the next neutrino to hit your detector is random because QM says it is.

But is it? How do we know that it wasn't determined at the moment of the big bang? And that if we only knew enough, we could predict it? Or that even if we could never know enough, it's still only random by virtue of our lack of knowledge?

This gets into one of the great disputes of twentieth century physics. Is it possible there are "hidden variables," aspects of reality that we just don't know, that, if we knew them, would enable us to see the determinism lurking behind even quantum probabilities? I'm no expert on this stuff, but I understand that Bell's theorem rules out hidden variables. At least up to the limits of our present knowledge of physics.

There's a lot of physics that we don't know. We can't merge quantum physics with general relativity. We don't know what dark matter or dark energy are. There are revolutions waiting to be explicated by geniuses not yet born. It's possible that there is no such thing as randomness; and that probability is simply a measure of our ignorance of the causes of events.

https://en.wikipedia.org/wiki/Hidden-variable_theory

https://en.wikipedia.org/wiki/Bell%27s_theorem
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What, conceptually, is probability? What is something being likely to happen?

Deep issue, and you've got some good answers already. I'll just introduce the theme of the theoretically fair coin. This is an ideal coin. It's the prototypical RNG (random number generator). Each time it's activated, it is just as likely to gives heads as tails (or 1 as 0, if you prefer to think of a random bit.) It has no memory, no anticipation. It's doesn't matter that the last 1000 flips were heads. The next flip is just as likely to be heads as tails. I don't know enough about modern physics to say whether anything in nature corresponds with it, but we seem to co-imagine it pretty well. Do we just have an intuition about RNGs that's hard to say more about?
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The next flip is just as likely to be heads as tails.

Only by virtue of our ignorance of the physical determinants of the outcome. Else you must not believe in physics at the macroscopic level.

See for example

and

https://phys.org/news/2009-10-tails-key-variables.html
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I see probability as a count of all possible scenarios in respect to it's results. For example, suppose a coin is tossed. That coin, thrown randomly, has an infinite number of ways it can fall considering it's speed, position, angle at which it lands, etc. However probability can simplify that to 50/50 because both sides have the same infinite number of scenarios that can lead to it. They cancel out, simply put and allow a simplified understanding of what is technically something of infinite possibilities.

Because of its nature, there needs to be either a very concrete understanding of how things can lead to the results or a large amount of trials to be able to apply the concept of probability to something. Going further with the coin example, a concrete understanding would be like what was mentioned previously, where the probability of 50/50 was reached deductively. The other way is to inductively find it by for example taking 100000 trials and finding something close to the 50/50 ratio.

But in the end it is the same. It is a simplified way for us to count the infinite possibilities for something.
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Only by virtue of our ignorance of the physical determinants of the outcome. Else you must not believe in physics at the macroscopic level.

Nature aside, what I'm trying to get at is the ideal, mathematical model. Specifically, we can think of a Bernoulli variable with p = 0.5. But even this is a formalism that aims at an notion. Actual coins are used as 'approximations' of the ideal coin, just as we settle for PRNGs.
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Nature aside, what I'm trying to get at is the ideal, mathematical model. Specifically, we can think of a Bernoulli variable with p = 0.5. But even this is a formalism that aims at an notion. Actual coins are used as 'approximations' of the ideal coin, just as we settle for PRNGs.

You're right, I reread your post. You're talking about a theoretically random coin, and asking if such a thing could exist or if it's only an object of our intuitions. So that's the right question, if I understood you correctly.
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Yes, I was trying to bring this intuition of the ideal coin into the conversation. Did we evolve so that this intuition was available? And: how does it connect to the larger conversation? Whether nature offers anything comparable is of course another important issue. (Does God play dice? Etc.)
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What, conceptually, is probability?
The word "probability" was derived from the concept of a provable postulate or prediction. An un-provable prediction is an opinion with no testable grounds for belief. Such prophecies must be taken on faith in the soothsayer, not on any objective evidence pointing to a normal future state. Hence, the prediction may rely on the small possibility of abnormal events (black swans) or miracles (divine intervention).

Conceptually, a highly probable event doesn't have to be taken on faith, it's almost certain. But the lower the probability, the more faith is required for belief. For example, the likelihood of a tornado hitting my home may seem remote, but if the weather forecaster has a good record of reliability, you'd be wise to take his word for it, and prepare to take shelter.

Mathematical probability is a numerical evaluation of the odds that the future state predicted can be tested and found true. As a practical method, probability theory derives its power from the stability of the "normal" Bell Curve behavior of large numbers of relevant objects or trials. In other words, we predict the future based upon past experience. But, the flaw in that theory is the small probability of Black Swans, that don't conform to the Norm. Some people interpret such rare events as miracles, because their minuscule probability is hard to calculate. Nevertheless, we can still retro-compute the probability after the fact ; after the evidence has been found ; after the event has "come to pass". :smile:

The law of large numbers is a principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes.
https://whatis.techtarget.com/definition/law-of-large-numbers

Black Swan theory : 2. The non-computability of the probability of the consequential rare events using scientific methods (owing to the very nature of small probabilities).
https://en.wikipedia.org/wiki/Black_swan_theory

True Prophets : Make predictions that “come to pass” (Jeremiah 28:9)
When a prophet speaks in the name of the Lord, if the word does not come to pass or come true, that is a word that the Lord has not spoken; the prophet has spoken it presumptuously. You need not be afraid of him. (Deuteronomy 18:20-22)
Beloved, do not believe every spirit, but test the spirits to see whether they are from God, for many false prophets have gone out into the world. (1 John 4:1-6)
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Probability is uncertainty.

If x + 1 = 1, then I'm absolutely certain, I have no doubts whatsoever, that x = 0. Probability has no role here.

On the other hand, if 5 < x + 1 < 10 where x is an integer, x could be 5 or 6 or 7 or 8. I'm uncertain of x's value. Probability enters the picture and without any further information, the probability that x = 5 is 1/4 = 25%. The probability that x >= 6 is 2/4 = 1/2 = 50%.
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I propose thinking of probability a bit more abstractly.

Probability fundamentally is a measure; when you assign a probability value, you're assigning a measure. The assignment of numerical probabilities is explicitly a claim that the measure of probability of events scales to the measure of the assigned numbers. To claim that A is twice as likely as B is to claim that P(A)=2P(B); to claim that you cannot say A is twice as likely as B is to claim that you cannot say P(A)=2P(B).
I mean, we assume that if an event has probability of 99,9% of happening, it means that if we simulate the conditions, each 1000 times the event would occur next to 999 times. But that's not a fact, since nothing really prohibits the complement of the event, with probability of 0,1% of keep continuously occurring through time, while the first event, with almost 100% of probability never happens.
I disagree with this assessment. To claim that P(A)=99.9% is to claim that you can assign a probability measure of A happening to the value 0.999. Since the numerical assignments presume the numerical scales map to the probabilities, and since 0.999 is 999 times greater than 0.001, the statement P(A)=99.9% is equivalent to the claim that A is 999 times as likely to occur as it is to not occur. That is what P(A)=99.9% means.

That does not imply that if you repeat the event 1000 times, then A should occur nearly 999 times (this being an application of the Law of Large Numbers). At best it implies that A is likely to occur nearly 999 times. Nor does this imply that if you repeat the event 1000 times, A would occur at least once; at best, it implies that it is likely to occur at least once. All P(A)=99.9% implies is that there exists a meaningful measure such that you can say things like A is 999 times more likely to occur than it is to not occur. In fact, as @Aryamoy Mitra did earlier, we can use probability to describe exactly how unlikely it is that A never occurs when the event happens 1000 times.
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Probability means that out of a number of events each has the same likelihood of happening. But how can we say they are all equally likely? How can we know that?
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Probability means that out of a number of events each has the same likelihood of happening

You have a container in which there are ten black balls and five white balls. You reach in a pluck out a ball. What is the probability it is black? White?
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You have a container in which there are ten black balls and five white balls. You reach in a pluck out a ball. What is the probability it is black? White?

black = 2/3
white = 1/3
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Well, it's an awkward question, but, what in fact is probability?
Humans have an advantage over most animals, in that we can imagine the near future, and prepare to make our next move, before the future actually arrives. Most animals deal with unexpected events with automatic knee-jerk reflexes. Which serves them well, in their narrow niche of the tooth & claw jungle. But humans have created a variety of artificial niches to suit diverse specialized needs and preferences. Consequently, our "asphalt jungle" is even more complex & chaotic, and rapidly changing, than the natural habitat of other animals.

That may be why we were forced to supplement our basic animal survival instincts, with formal methods for more accurately predicting the moving targets of the future. Ancient prophecy was merely educated guessing, based on direct experience from past events and trends. But humans also learned to create abstract mathematical models of how the world works. And Probability Theory eventually emerged, ironically from Game Theory, based on long experience with gambling competitions, to give those-in-the-know an advantage over other players. For example, a card game is an abstract simulation of real-world social situations. If you can "count cards" you will have a better idea of what hand your opponent is holding, and what his next move might be. Hence, when such unknowns can be reduced to number values they can be manipulated more quickly & accurately than the nebulous social values of human communities : e.g is he bluffing?.

Therefore, what we now call "Probability" is essentially a formalized form of intuition or foresight. It allows us to calculate what is normally-to-be-expected in a well-defined situation. Hence, It gives us an edge in dealing with the unnatural exigencies of the complicated civilized world of cunning thinking animals, and with the unfamiliar uncertainties of the natural world. Probability Theory is "in fact" a new tool, like teeth & claws, for humans to use in the high-stakes game of survival. Unfortunately, Probability is still not a perfect form of Prognostication. :brow:

There are three major types of probabilities:
Theoretical Probability.
Experimental Probability.
Axiomatic Probability.
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Humans have an advantage over most animals, in that we can imagine the near future, and prepare to make our next move, before the future actually arrives.

Like a cat badgering you till you open a can of cat food for it? Cats most definitely imagine the near future. "Rub my tummy, human." "Feed me, human."
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Cats most definitely imagine the near future.
The human advantage over cats is in the degree & detail of its imagery -- including abstract models of Probability. I assume that cats have an instinctive sense of future prospects, but the theory of Probability goes beyond the innate dispositions that humans share with cats, into ideal realms where the cat food doesn't require human servants with hands & can openers. What if cat food just grew on trees -- what are the chances? :joke:
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Probability is a measure of information about a system. For two bettors, odds differ for each depending on what each knows. The Monty Hall problem makes this reasonably clear.
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Cats most definitely imagine the near future.
The human advantage over cats is in the degree & detail of its imagery -- including abstract models of Probability. I assume that cats have an instinctive sense of near- future prospects, but the theory of Probability goes beyond the innate dispositions that humans share with cats, into ideal realms where the cat food doesn't require human servants with hands & can openers. What if cat food just grew on trees -- what are the chances? :joke:
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Denis / All

As I have mentioned on my thread in the Philosophy of Science forum, probabilities are an acknowledgment of no known cause for a range of outcomes, and therefore their function is to provide a description in the absence of an explanation.

In providing that description, I agree with T Wood that
Probability is a measure of information about a system.

Determinist theory, and the maths involved in the traditional Laws of Physics & Chemistry, (rather than the principles emerging from QM), says that for every action from a precise start point, there can only be one inevitable outcome - and if you were to accept that premise for all aspects of existence, then everything in existence would be truly inevitable and we would all be acting-out a fixed script.

The trouble is that this theory contradicts our living experiences, and requires us to effectively deny both the reality of our lives, and the experiences of every person who has ever lived, simply to uphold a doctrine - which may or may not be true. That is a very big ask.

However, determinists will point out that there is a difference between our limited knowledge and therefore our ability to predict outcomes vs the underlying inevitability of existence. There is an important difference in that respect, but it doesn't deny the possibility of truly random events instead of hidden causes/variables.

The Laws of Physics demand one outcome - not multiple outcomes from a given start point, (called randomness) - because that is the basis of their explanations. So when multiple outcomes arise, there is either a missing factor, or a lack of determinism.

In trying to resolve the unknown factors behind the differences in outcome, people may try to assume that the gap simply reflects known factors that are not being monitored. But that is an assumption, and where there is no factor that can logically be applied to resolve a scenario, then such a presumption seems to be at odds with the evidence. It is perfectly valid to look for missing factors, but it is also valid to consider that there may be genuinely non-determinist (spontaneous or random) factors in existence.

The more narrow the range of potential outcomes, the more likely it would seem that a hidden mechanism is at work, which has not yet been identified - but conversely, a very broad range of outcomes, without any discernible pattern might also be evidence for a lack of determinism.

As has been pointed out, probabilities, like odds in a lottery, provide a generalised description rather than a firm prediction. The most extreme and unusual outcome could occur next time around. But the converse is also true, probabilities indicate likelihood, and so the chances of the same extreme outcome occurring a second or third time would be extremely remote if the statistical model were originally correct.

That is the dilemma when trying to apply probabilities to the origin of the first living cell. Multiple examples of every single protein would be needed for nature to experiment and evolve. When the odds of a single protein occurring by chance are one in '10 to the power 366' (ie. more options than there are atoms in the universe), then yes the likelihood of a 2nd example occurring by chance are pretty well nil. Yet life emerged.

In other words, the problems are not confined to randomness, but also examples of co-ordination that break the scientific models. So I don't agree with fishfry that

We are hard pressed to give even a single example of an ontologically random event. Most people will fall back on quantum events. The low-order bit of the femtosecond timestamp of the next neutrino to hit your detector is random because QM says it is.

The traditional main source of examples for a lack of determinism arise from Thought and consciousness. But as we have seen, the origin of proteins, plus the growing number of examples from the activities of molecules within living cells, point to other equally challenging issues. For instance, (as again discussed in the Philosophy of Science forum), certain molecules display characteristics of problem solving which defy the logic of the Laws of Physics which should apply to them.

In terms of randomness there are also many examples from other disciplines, such as cosmology including the broad theory of origin and the Big Bang. The evidence of the accelerating expansion of the universe either says that the 'Big Bang - Big Crunch' explanation must have had a start point 13.7 billion years ago (representing spontaneity without cause), or had a change to a previously eternal sequence - requiring a change that had to be either spontaneous or random - ie. non-deterministic. (I confess that I got that principle from Finipolscie's books).

The problem with QM is that we can't see what's happening at that level of existence, and can only describe what is observed. The extensive use of probabilities in QM is a way to try to bring the perceived randomness of the observations into the deterministic fold - but they still represent a description rather than an explanation, because they are basically an admission that we don't have a cause to explain the different outcomes.
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Well, it's an awkward question, but, what in fact is probability? I mean, we assume that if an event has probability of 99,9% of happening, it means that if we simulate the conditions, each 1000 times the event would occur next to 999 times. But that's not a fact, since nothing really prohibits the complement of the event, with probability of 0,1% of keep continuously occurring through time, while the first event, with almost 100% of probability never happens.
Anyone could argue that this is not likely, or if it happens, if you repeat the experiment, it would probably not happen again. But those two arguments just use again the definition of probability without explaining it.
I could add that saying "Means that something is more or less likely" just change the word probable for likely.
I know that in real life events like the first described are not usual, but mathematically it's not impossible and it's just a scenario to the main question: What, conceptually, is probability? What is something being likely to happen?

Here’s my take: Probability is a quantified occurrence of a possible event under limited conditions.

The difference between probability - a mathematical structure - and real life events is the distribution of potential energy into attention and effort, or affect.

The important point you make is ‘if we simulate the conditions’. So, if an event has a 99.9% probability of happening (under certain conditions), then theoretically we could manipulate these conditions locally in such a way that its complement will always occur, with surprising accuracy. The more precisely controlled and controllable the conditions under which this probability is calculated, the more play we have.
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# Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$

I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a series of algebraic manipulations that can lead from $\sum\limits_{i=0}^n \binom{n}{i}$ to $2^n$.

• is the proof you looking for using $(1+1)^n=2^n$? – Arjang Jan 24 '11 at 4:30
• Well, no. That one I was also aware of. It's more of a curiosity if there's any direct method to go from the summation to $2^n$. – JSchlather Jan 24 '11 at 4:40
• One should not think of the algebraic and combinatorial proofs as different. There is a straightforward dictionary between algebra and combinatorics in these cases (and it is given by taking generating functions). – Qiaochu Yuan Jan 24 '11 at 9:12
• Zeilberger's algorithm might do it - it's a useful tool for this kind of problem in general (sum from $-\infty$ to $\infty$ of a hypergeometric with finite support). – Peter Taylor Jan 24 '11 at 9:29
• @PeterTamaroff It was a sort of silly question that I asked over a year ago because I had always thought of binomial coefficients being defined as $n!/(k!(n-k)!)$ and was always curious how one could go from this sum of factorials divided by other factorials to $2^n$. When I had said direct, I had meant something along the lines of proving the binomial recurrence by moving factorials around. It's not a very clear question in retrospect. But people seemed to get the basic idea of what I wanted. – JSchlather Apr 27 '12 at 19:23

Here's one. Let $g(n) = \sum \limits_{i=0}^n \binom{n}{i}$. Then

$$g(n+1) - g(n) = \sum_{i=0}^{n+1} \binom{n+1}{i} - \sum_{i=0}^n \binom{n}{i} = \sum_{i=0}^{n+1} \left(\binom{n+1}{i} - \binom{n}{i}\right) = \sum_{i=0}^{n+1} \binom{n}{i-1}$$ $$= \sum_{i=0}^n \binom{n}{i} = g(n).$$ Here, we use the fact that $\binom{n}{n+1} = \binom{n}{-1} = 0$, as well as the binomial recurrence $\binom{n+1}{i} = \binom{n}{i} + \binom{n}{i-1}$.

Thus we have $g(n+1) = 2g(n)$, with $g(0) = 1$. Since $g(n)$ doubles each time $n$ is incremented by 1, we must have $$g(n) = \sum_{i=0}^n \binom{n}{i} = 2^n.$$

• This is more or less the same proof one would do to show that $(a+b)^n$ is what it is... – Mariano Suárez-Álvarez Jan 24 '11 at 5:40
• @Mariano: Good observation. In fact, see the proof of Identity 1 in this paper: math.pugetsound.edu/~mspivey/CombSum.pdf – Mike Spivey Jan 24 '11 at 5:57
• Very nice and this proof seems to be analogous to what picakhu did as well. – JSchlather Jan 24 '11 at 5:57

Simply use the binomial formula.

$$(a + b)^n = \sum_{k=0}^n {n \choose k} a^k b^{n - k}$$

With $a = b = 1$ you have your result.

Well, here is one.

$$\sum_{i=0}^n \binom{n}{i}=2^n$$ $$\sum_{i=0}^n \binom{n}{i}+\sum_{i=0}^n \binom{n}{i}=2^{n+1}$$ $$\binom{n}{0}+\left [ \binom{n}{0}+\binom{n}{1} \right ]+...+\left [ \binom{n}{n-1}+\binom{n}{n}\right ]+\binom{n}{n}=2^{n+1}$$ $$\sum_{i=0}^{n+1} \binom{n+1}{i}=2^{n+1}$$

• So you're using induction? And I assume that in your last step it should be $\sum_{i=0}^{n+1}\binom{n+1}{i}$? – JSchlather Jan 24 '11 at 4:41
• Yup, sorry about that. – picakhu Jan 24 '11 at 4:43
• $\sum_{i=0}^0 \binom{0}{i} = 2^n$ – zapyourtumor Mar 24 '18 at 3:02

You could use exponential generating functions to prove this identity. $$\begin{eqnarray}\sum_{n\ge0}2^n\frac{x^n}{n!}&=&\sum_{n\ge0}\frac{(2x)^n}{n!}\\&=&e^{2x}\\&=&e^xe^x\\&=&\sum_{i\ge0}\frac{x^i}{i!}\sum_{j\ge0}\frac{x^j}{j!}\\&=&\sum_{n\ge0}\sum_{i=0}^{n}\frac{x^i}{i!}\frac{x^{n-i}}{(n-i)!}\\&=&\sum_{n\ge0}\sum_{i=0}^n\binom{n}{i}\frac{x^n}{n!}\end{eqnarray}$$ Now, comparing the coefficients of $x^n$ for ${n\ge0}$ gives the identity.

Using the binomial theorem, we find:

$2^n = (1 + 1)^n = \sum\limits_{k=0}^{n} \binom{n}{k} 1^k*1^{n-k} = \sum\limits_{k=0}^{n} \binom{n}{k}$


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# Are the boundary conditions purely a consequence of Maxwell's equations?

The boundary conditions, namely

were all these, realized only by looking at Maxwell's equations? Or is there a physical reasoning behind them? For example, Why does the component of the electric field parallel to the surface of interface remain unaltered? I also read that the reason light bends when it passes through another medium is because only the normal component gets altered and the horizontal component remains the same(whereas the velocity gets altered because of the other electrons in the material that are driven by the source and produce a separate wave with a different phase and the superposition of these two waves seem to alter the speed of light in a medium $$^\dagger$$).

My question in short is, What would be my answer,if someone asked me, to explain the boundary conditions, without equations*.

*If that's purely based on equations, please ignore "without equations", but there's got to be something that's physically occurring which led us to create a model, right?.

$$^\dagger$$Is that right?


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# Minmax problem for polygons

Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ vertices. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles which vertices are some vertices of $P$. I want to find an explicit formula for the function $$\Phi(n)=\inf\limits_{P\in \text{Pol}_n}\max\limits_{T\in \text{Tr}(p)}\frac{\text{area}(T)}{\text{area}(P)}$$ It is not hard to prove that $\Phi(3)=1$, $\Phi(4)=1/2$. For $n\geq 5$ we have an estimation $\Phi(n)\geq 1/(n-2)$.

Here you can find some attempts to solve it. Any ideas are appreciated!

-
Somewhat related question: mathoverflow.net/questions/34865/… – Gerry Myerson Mar 11 '12 at 1:18
An algorithm for finding a maximal triangle is given in cs.bgu.ac.il/~sityon/soda06.pdf. I don't know whether the paper, or its references, contains any theoretical results on the size of said triangle. – Gerry Myerson Mar 11 '12 at 1:21