When To Ignore Changes In Pressure When Computing Flow Energy For Gases?

Question:
In reference to this example problem (only partially shown), I need to understand why for $\Delta h$ the change in enthalpy, which is $\Delta P/\rho + \Delta u$, the change in pressure is ignored in the solution even though it is considerably large. Could somebody please explain this to me?
Answer: Starting Point
The starting point of the difference form of the steady state energy balance equation is below.
$$ \dot{q} - \dot{w} = \dot{m}\Delta\left(\tilde{U} + \frac{p}{\rho} + \frac{v^2}{2} + gz\right) $$
In this, $\dot{q}$ is the heat flow (J/s - negative leaving the system), $\dot{w}$ is the Clausius form of work (J/s - positive when done by the system), $\dot{m}$ is the mass flow (kg/s), $\tilde{U}$ is the specific internal energy (J/kg), $p$ is pressure (Pa), $\rho$ is density (kg/m$^3$), $v$ is velocity (m/s), $g$ is gravity (m$^2$/s), and $z$ is height (m).
For any system, the definition of specific enthalpy is $\tilde{H} \equiv \tilde{U} + p\tilde{V} = \tilde{U} + \frac{p}{\rho}$. The expression becomes as below.
$$ \dot{q} - \dot{w} = \dot{m}\Delta\left(\tilde{H} + \frac{v^2}{2} + gz\right) $$
System
Take the control volume as the fluid inside the compression flow. Allow adiabatic so that $\dot{q} = 0$. Redefine $w$ as purely shaft work because the system (fluid) does not expand or contract as a whole (see below about incompressible fluids or ideal gases).
The system is horizontal in $z$. This is identical to stating categorically that $\Delta z \equiv 0$. The somewhat questionable approach to propose to neglect change in potential energy misses the clarity that zero change in height is a fact of the choice of control volume and not a (randomly applied) assumption.
The velocity of the fluid at any point is found from volumetric flow $\dot{V}$ and area $A$.
$$\dot{m} = \dot{V}\rho = vA\rho$$
Incompressible Fluid
For an incompressible fluid AND ONLY FOR THIS CASE, we can use the conservation of energy expression to obtain this expression at any point in the flow.
$$ v = v_i \frac{A_i}{A} $$
The statement that we "neglect" changes in kinetic energy can therefore only mean that we correspondingly assume that the area changes are negligible. Otherwise, the full representation of the energy balance is as below.
$$ - \dot{w}_s = \dot{m}\Delta\tilde{H} +\frac{\dot{m}v_i^2}{2} \left(\frac{A_i^2}{A_f^2} - 1 \right) $$
For a system with constant specific heat capacity $\tilde{C}_p$ (J/kg K), this can be written as below using change in temperature $\Delta T = T_f - T_i$ ($^o$K or $^o$C).
$$ - \dot{w}_s = \dot{m}\tilde{C}_p\Delta T + \frac{\dot{m}v_i^2}{2} \left(\frac{A_i^2}{A_f^2} - 1 \right) $$
Ideal Gas
The opposing case to an incompressible fluid is the ideal gas. In this case, we obtain the expression below using molar mass $M$ (kg/mol) and gas law constant $R$ (J/mol K).
$$ \dot{m} = \frac{\dot{n}}{M} = \frac{p}{MRT}vA $$
This leads to a modification of the velocity equation as
$$ v = v_i \frac{\beta_i A_i}{\beta A} $$
with $\beta = p/MRT$. For a system with negligible pressure drop from inlet to outlet, the energy balance becomes as below for an ideal gas.
$$ - \dot{w}_s = \dot{m}\tilde{C}_p\Delta T + \frac{\dot{m}v_i^2}{2} \left(\frac{\beta_i^2 A_i^2}{\beta_f^2 A_f^2} - 1 \right) $$
Summary
The solution states that potential energy change is negligible. In fact, $\Delta z = 0$ is an absolute truth, not a required assumption.
The solution states that kinetic energy change is negligible. This can be a gross mistake depending on the type of fluid (incompressible or ideal gas), the extent of the throttling ratio $A_f/A_i$, and for an ideal gas the ratio of temperature and pressure drops through the system.
Answer: When calculating flow energy, you should always take pressure or temperature changes into account as they are linked by ideal gas law ($PV = nRT$). In this case, calculating change in enthalpy due to temperature change is equivalent to calculating it using pressure change.
Comment: Hi Alexandre. Thank you for your reply. Do you know a source that could support what you've said (that dU and dP represent the same value for a gas)?
Comment: I have linked a table from Wikipedia. I am not saying dU and dP represent the same value. Don't get confused with the $C_v$ and $C_p$ which are linked with the following relation :
$C_p = C_v + R$

From the general equation $H=U+pV$, using $Q=0$  and $C_p$-$C_v$ relation you should be able to show that $\Delta h = C_p*\Delta T$.


Switch console output to RS232

Question: How can I switch temporally Linux ASCII console to RS232 port build-in computer. I mean about Linux system working as server, without running any graphical tools on it (X11, bootsplash etc).
But I want to do it temporally just "on demand". In ordinary usage, the console should be normal VGA monitor. Is it possible?
Comment: Do you really mean "the console" or are you simply trying to attach a terminal via a serial port? In Unix, the console is one of many ttys, but it hapens to be the one where the pre-boot/boot functionality is configured via, and where some messages are written. Do you need the latter?
Answer: Possible? Yes.
Temporarily? Too much of a hassle.
It's all well documented in the kernel documentation, and in modern distributions it should be sufficient to edit <code>/etc/inittab</code> in order to have a terminal listening on your serial port - which is what I assume you want to do.
If you just want the console messages displayed, edit your boot options accordingy.
Comment: OK. I place something like `0:respawn:/sbin/agetty -L ttyS0 -H MEDIA 9600 vt100` but the problem is that /etc/inittab is no longer valid because systemctl is used. So what and where I have to place?
Comment: This would be a different question, and could be solved with minimal research effort, but here goes: http://bit.ly/13Uz2zi

Switch console output to RS232


hp printer install, start a new, how

Question: oke so i have new HP printer, this is what happend (so far)
HPLIP 3.17.7 was pre-installed
tried to install 3.17.11, but there were dependency issues so i left it for what is was as it wasn't really necessairy.
i unpacked, linked to wifi and installed via HPLIP toolbox (first via USB, and then it was later added via WiFi)
all worked fine (then) except:
There were 2 printers (slightly differently named) and one fax installed under devices, While only 1 printer and 1 fax under HPLIP toolbox.
I tried to remove them and re-install them (the two printers just didn't seem right). but this morning i found there was only one printer left, the two under hplip toolbox were gone.
I also found out the printer didn't wake-up when an job was sent but it did print when i manually activated the printer.
i tried to re-install the printer via hplip toolbox but forgot to take out the usb so things were working, but via usb not wifi. In the mean time i also tried to scan.. which worked (mostly) but also via usb  ofcourse.
Now i took the usb away i found that there is just one printer under devices but i can't seem to have hplip to find the printers (while the setup runs normally) also the scanner is not found under xsane.
anyway i would love to start a new.
for that i believe i should:
reset the printer.
remove all "instances" of the printer in ubuntu.
i would like to actually install hplip 3.17.11
but the last 2 i have problems with:
as the last remaining printer comes back if i remove it even with the printer powered off. i tried in devices and in cups!
about hplip version i'm in doubt, hp tells me there is support for 17.10 64 bit added, but why does ubuntu 17.10 then come with 3.17.7 instead. also the printer itself doens't really need it.
the error i get when trying to install is the following.
<code>RUNNING PRE-INSTALL COMMANDS
----------------------------
OK

MISSING DEPENDENCIES
--------------------
Following dependencies are not installed. HPLIP will not work if all
REQUIRED dependencies are not installed and some of the HPLIP features
will not work if OPTIONAL dependencies are not installed.
Package-Name         Component            Required/Optional
libcrypto            network              REQUIRED
libnetsnmp-devel     network              REQUIRED
sane-devel           scan                 REQUIRED
pyqt4-dbus           gui_qt4              REQUIRED
cups-devel           base                 REQUIRED
reportlab            fax                  OPTIONAL
cups-image           base                 REQUIRED
python-notify        gui_qt4              OPTIONAL
libjpeg              base                 REQUIRED
dbus                 fax                  REQUIRED

Do you want to install these missing dependencies (y=yes*, n=no, q=quit) ? y

INSTALL MISSING REQUIRED DEPENDENCIES
-------------------------------------
note: Installation of dependencies requires an active internet connection.
warning: Missing REQUIRED dependency: cups-devel (CUPS devel- Common
Unix Printing System development files)
warning: Missing REQUIRED dependency: cups-image (CUPS image - CUPS image development files)
warning: Missing REQUIRED dependency: libjpeg (libjpeg - JPEG library)
warning: Missing REQUIRED dependency: libcrypto (libcrypto - OpenSSL cryptographic library)
warning: Missing REQUIRED dependency: libnetsnmp-devel (libnetsnmp-devel - SNMP networking library development files)
warning: Missing REQUIRED dependency: pyqt4-dbus (PyQt 4 DBus - DBus Support for PyQt4)
warning: Missing REQUIRED dependency: dbus (DBus - Message bus system)
warning: Missing REQUIRED dependency: sane-devel (SANE - Scanning library development files)

INSTALL MISSING OPTIONAL DEPENDENCIES
-------------------------------------
note: Installation of dependencies requires an active internet connection.
warning: Missing OPTIONAL dependency for option 'gui_qt4': python-notify (Python libnotify - Python bindings for the libnotify Desktop notifications)
warning: Missing OPTIONAL dependency for option 'fax': reportlab
(Reportlab - PDF library for Python)

CHECKING FOR NETWORK CONNECTION
-------------------------------
Network connection present.

RUNNING PRE-PACKAGE COMMANDS
----------------------------
sudo dpkg --configure -a (Pre-depend step 1)
sudo apt-get install --yes --force-yes -f (Pre-depend step 2)
sudo apt-get update (Pre-depend step 3)
OK

DEPENDENCY AND CONFLICT RESOLUTION
----------------------------------
Running 'sudo apt-get install --assume-yes libcups2-dev'
Please wait, this may take several minutes...
error: Command failed. Re-try #1...
Running 'sudo apt-get install --assume-yes libcups2-dev'
Please wait, this may take several minutes...
error: Command failed. Re-try #2...
Running 'sudo apt-get install --assume-yes libcups2-dev'
Please wait, this may take several minutes...
error: Command failed. Re-try #3...
Running 'sudo apt-get install --assume-yes libcups2-dev'
Please wait, this may take several minutes...
error: Package install command failed with error code 100
Would you like to retry installing the missing package(s) (y=yes*, n=no, q=quit) ?
</code>
additionally
<code># hp-scan -g

HP Linux Imaging and Printing System (ver. 3.17.7)
Scan Utility ver. 2.2

Copyright (c) 2001-15 HP Development Company, LP
This software comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome to distribute it
under certain conditions. See COPYING file for more details.

hp-scan[3467]: debug: getDeviceUri(None, None, ['hpaio'], {'scan- type': (<built-in function gt>, 0)}, , True)
hp-scan[3467]: debug: Mode=0
error: No device selected/specified or that supports this functionality.
</code>
anyways, i would GREATLY appreciate help on all of this.
Answer: Well for now i seem to have solved this issue by:
1) reset printer to factory setting
2) updated ubuntu to newest version (still 17.10) but a fresh install
<code># sudo do-release-upgrade -d

hplip:
  Installed: 3.17.10+repack0-4
  Candidate: 3.17.10+repack0-4
  Version table:
 *** 3.17.10+repack0-4 500
        500 http://us.archive.ubuntu.com/ubuntu bionic/main amd64 Packages
        500 http://archive.ubuntu.com/ubuntu bionic/main amd64 Packages
        100 /var/lib/dpkg/status
</code>
Comment: good work; well done

hp printer install, start a new, how


What causes seals to fail when canning?

Question: My husband made and canned pickles 1 week ago. Tonight he noticed the seals were popped on all but one jar. One other time the same thing happened when he canned green beans. What would be the cause of this?
Comment: It's unclear if we're talking about boiling water or pressure canning here - I'd expect boiling for the pickles and pressure for the green beans (since they're not acidic enough)? I think the answers are mostly the same either way though.
Answer: Aside from the notes in SAJ14SAJ's answer, if he didn't run the canning process long enough, there may have been residual bacteria (or spores) that remained active.
The failed cannings I've seen due to insufficient heat took exactly one week to pop their lids.
Answer: You should not re-use lids, so they should come in good condition, although you can give them a visual inspection for dents, dings, and missing sealant before using them.
However, it sounds like perhaps your jars are not being properly processed to develop their seals if so many are failing.  The University of Missouri Extension provides this list of things that might have gone wrong:

Failure to read and follow instructions on package for using bands and lids.
Leaving wrong amount of head space when filling jars.
Not removing particles of food, seeds, seasonings, or pulp of fruit from the top of jar before putting the lid on. Also, watch for threads from the cloth that is used to clean the rim of the jar.
Screwing band too tight can cause lid to buckle.
Not screwing band tight enough.
Insufficient heat to seal the lid.
Failure to exhaust (vent) steam - pressure canner for a full 10 minutes.
Letting pressure in canner fluctuate.
Reusing jars in which commercial food is sold.
Using warped, rusty, worn bands, or old commercial caps instead of home canning brands that are in good condition.
Reusing lids. The lid of a two - piece metal cap should not be used more than once.
Comment: The [NCHFP's page](http://nchfp.uga.edu/how/general/cannedfoodproblems.html) is of course also good. It manages to mention a few different things despite missing a lot of them: defects in the jar itself (chips in the glass), tipping/inverting jars when removing them from the canner, and (when canning meat) getting fat on the rim.

What causes seals to fail when canning?


Important photos that was on my phone

Question: I had a Windows Phone 7, and it was stolen.
I thought my pics were backed to Google but i cant find them anywhere. Where does Windows Phone Backup pictures.
Comment: Can you check your OneDrive account.
Answer: Microsoft has its own cloud service, called Onedrive, which was also formerly known as SkyDrive during the Windows Phone 7 days. Every Windows Phone device. by default backs up pictures in the Camera Roll, Screenshots and Saved Pictures onto the cloud:

Visit onedrive.live.com and sign in with the Microsoft (Outlook, Live, Hotmail) account that you used on that particular phone, and navigate to a folder named Pictures, in the root directory of Onedrive. Under there, you are bound to see sub folders of the photos uploaded from all your Windows powered devices.
Answer: if you had Onedrive on your phone hopefuly theyre there. simply go to google and type OneDrive and sign in with your previous Microsoft account and password. hope you get them

Important photos that was on my phone


How to check the authenticity of the root cert of some CA?

Question: Verifying and trusting some root certificate of an unusual CA might present a problem. In fact, to download the root cert of this CA from its corresponding website one should verify and trust first the TLS cert presented by that website. Meaning trusting the root cert of the CA signing the website's TLS cert. It seems like a chicken-and-the-egg problem. How could one solve that ?
Answer: Already good answers. I will focus mine solely on the chicken and egg problem.
Egg: You are trying to validate a certificate, but the cert chains to a root that you have never seen before.
Chicken: To decide whether you should trust this CA, you look at who issued the root cert, but the issuer of a root CA cert is always ... itself, so we're back to the egg scenario.

Hence the name "root CA certs" for "root of trust". If you trust the root, then you trust the entire tree. If you don't trust the root, then you don't trust the entire tree. But how to determine if a root CA is trustworthy?
There is no automated way to determine whether a root CA is trustworthy, and that's kinda the point. Basically, you need to get to know the organization (ie the humans) who run that CA and decide whether they are trustworthy and doing their jobs properly. As an example, let's look at Mozilla's process for deciding whether a CA should be included in the truststores of Firefox and other products.
First, see: Mozilla's CA Certificate Program which overviews every detail of their process around the root trust store (huge kudos to them for making all this public!).
Next, we dive a little deeper into their CA Application Process. As you can see, there is a very rigorous and time-consuming process by which Mozilla determines whether a CA is "good enough" to be included in the Mozilla trust store.

Bottom-line: there is no over-the-internet way to determine whether a root CA is trustworthy (I guess you could look them up on a forum or something, but how do you know if you should trust those people? See: chicken-egg). You need to first have in-real-life trust in the people running the CA. The other answers detail the risks of adding a malicious root cert, so I won't go over that.
Let's assume the CA you're thinking of adding is run by your company / school / friend: you believe them to be honest (not issuing fake certs to bad-guys), and you think they are have good security practices to prevent their CA from being hacked (highly unlikely unless, at a minimum, they have spent thousands of dollars on Hardware Security Modules to store their CA private key, and firewalls to protect their admin interface), then go ahead and add it.
Otherwise, if this is some root cert that you stumbled across on the internet, you would need to reach out to the organization behind the CA by phone and go through your own version of the Mozilla check-list (which probably involves several months of your time and several plane tickets on both your part and the CA's part).

Bottom-bottom-line: Unless the root CA is your company / school and you are literally required to add it for your computer to work, *then don't*. Mozilla / Google / Microsoft / Apple, etc have very rigorous processes for vetting CAs, so if that CA was not included, there's probably a reason.
Answer: You should never ever trust a new root CA unless you really (and I mean really really) believe that it is fully trustable. Even then you should get the CA certificate in a trusted way, i.e. definitely not by downloading it from a site which certificate is issued by the same yet untrusted CA.
The main problem with explicitly added root CA is that any explicitly added CA is automatically trusted to issue a certificate for any domain. This means even if example.com has usually a certificate signed by InnocentCA the browser will also accept without any notice a certificate signed by MaliciousCA, if you have added MaliciousCA as trusted CA. This makes man in the middle attacks or other kinds of domain impersonation by MaliciousCA easy.
Even certificate pinning (built-in or with HPKP) will not help since pinning is disabled in browsers if the certificate is signed by an explicitly added CA, to make legal SSL interception as used in enterprise firewalls and many desktop antivirus possible.
Answer: First, you have to consider why this organization believes they should be issuing their own root CA and distributing it to you.
Does this certificate belong to your own organization?  Does your organization manage their own PKI and issue their own internal certificates, for internal boxes to communicate with each other?  This is a common scenario in industry or other larger organizations, and is a perfectly legitimate reason to internally distribute certificates from an internal server to internal clients.  At that point you're discussing obtaining the certificates via a private network, where your organization presumably has security and controls on the distribution of root certificates.  Trusting such a server and certificate should pose little problem to you as a member of the organization.
If this is from some external site (perhaps one that doesn't understand certificate authorities and built their own PKI instead of paying for a certificate,) and is now asking their clients to trust their root CA, then no.  There is no reason to trust their certificate as a root CA.  So if this is a connection to a vendor's site, ask them to provision their server with a properly signed certificate from a well-known trusted root CA.  You're paying them for the service, not to introduce security vulnerabilities.  There is no reason they should put your systems at risk to use theirs.
If they are unwilling or unable to cooperate, you have to question their understanding of PKI and perhaps their competence in security in general.  Consider finding an alternate vendor who poses less of a risk.  If no alternative is possible and you have not choice but to use their services, you should act to minimize the risk to your organization.  For example, you could stand up your own proxy server that trusts their certificate and would connect to their server; your clients would only use and trust your proxy to access the vendor's services.  That would limit your risk to a single proxy system.
If you do have to accept their root CA on your client systems, be sure to import their certificate granting it only the minimum privileges.  When importing a root CA certificate, most systems will prompt you with a series of check-boxes: "trust this certificate to identify web sites?" "trust this certificate for digital signatures?" "trust this certificate for software updates?"  Be sure to only grant it the appropriate level of trust.
Answer: It's all to do with trust.
The root CA certificates of many come prelaoded within operating systems and some browsers. You may not even know this, but that's because these manufacturers have trusted them on your behalf (you can always remove any). If the CA site you are visiting does not have the root installed then you will receive an error - at this point you either trust the certificate and proceed so you can get the root to download, or you stop.
If you download and install a root CA then you are solely taking that trust on. If you do not trust the CA then do not install it's root certificate. If you are unsure whether to trust the CA then asking around is a good start, but that said if the root is not installed on other client machines then they will get a certificate error when they try to use the client certificate in the chain anyway. This therefore in your example means that you either chose to trust the error to proceed downloading the root certificate for that site, or you did not - as you would have received an error without the root certificate in the first place.
Answer: You can copy the thumbprint(or fingerprint whatever) of the Root CA's certificate, then google it, if it is authentic, it will show results of the CA's name exactly as yours.
For detail, assume you are using Chrome browser, you enter your target https site to verify,

Ctrl+Shift+I or COMMAND+Opt+I to open developer tool
Click "Security" tab
Click "View Certificate"

Click "Certification Path"
Double-Click Root Item
Click "Details" tab header
Scroll to "Thumbprint" and click it
Copy thumbprint showed in lower pane

Google the thumbprint (better be quoted)


Why does it show them arguing about TNE vs. TNA?

Question: In the Wolf of Wall Street, all of the founders are in Jordan's office and his dad, Max walks in and gets angry about how much theynhave spent on "dinner". Then Donnie says "It's okay, the IRA allows for T&A", then max corrects him saying its T&E and then Donnie argues with everyone that he did in fact say T&E, why does the movie show this?
link to scene in movie
Comment: I think it's supposed to be a comedic moment, with Donnie giving a freudian slip (TNA = "Tits 'n' Ass) which, if he made the same slip in front of an IRA agent, could mean big problems.
Answer: The four men are acting like children.  They're trying to make each other laugh, because dad is so angry and they're supposed to be taking the situation seriously.
Donnie says "T&A" (which is slang for t*ts and a**) with the intention of making his friends laugh, knowing that he can cover it up by saying that he actually said T&E (which is a business term that means travel & expenses).
The scene was included overall to show that the men think they can do anything they want without getting in trouble. The "T&A" joke included just as another example of the men acting immature, and laughing when they should be taking something seriously.
Answer: He says T&A because they spent the money on strippers and hookers. I mean escorts, like the real life person whom the film is based on was known to do. He literally means he spent it on Tits and Ass. The father knows that EJ entertainment is some sort of escorts service. Its a flippant reply to Max, who as their accountant is already mad about the mess they are putting in his hands. And Donnie continues trying to bullshit his way through it. They argue because Max cares and Donnie doesn't.
It is shown that way to illustrate to the audience how ridiculous and irresponsible they are with money, squandering it away. They don't care cause they know they can get away with it and someone like Max will fix it. It's just another sequence building up the characters as the immature jackasses that they are. And because the movie is a slightly dramatized and exaggerated retelling of an autobiography, it's likely that a similar conversation happened.


Help Lost Bitcoin phase

Question: I was using a Bitcoin wallet called "Bread" then transfer to another wallet called "Bitcoin".  The Bitcoin app was deleted from my IPhone 7.  The password word phase I wrote down was the same as the "Bread" app.  It that possible? Or did I write down my old password thinking I was in my new "Bitcoin" app.  I have tried download the Bitcoin app again but unable to restore the account.
Thank you, John
Answer: Though it is possible that both wallets generated the same mnemonic seed phrase (which is used to derive private keys), it is extremely unlikely. The chance of it being the same is 1 in 2^256, which is a very small number. This is obviously assuming no bugs in the code generating the key.
Restoring the account is only possible if you still have the mnemonic seed phrase. If you don't, then I'm afraid you won't be able to access the bitcoin until you find the seed.
Note that the mnemonic seed phrase is different than a user-supplied password. A mnemonic seed phrase will derive addresses, whereas a password will encrypt (lock) the wallet. So if you use the seed phrase to recreate the wallet on another device, you will not need the password to access the funds on the second device. That said, in some cases, a user-supplied password may also be used as part of the seed phrase (a '13th word or '25th word'). So you'll need to determine the nature of your password in order to recreate your wallet.
Answer: If you re-install the wallet app and re-enter the mnemonic seed, you should be okay, though your question is slightly ambiguous in wording.

Help Lost Bitcoin phase


Groups with many non-conjugate but orbit equivalent actions

Question: Which countable discrete groups (apart from the infinite amenable ones) admit uncountably many mutually non-conjugate free ergodic probability measure preserving actions that are all mutually orbit equivalent?
Comment: Many. Are you expecting the answer to provide a classification (don't) or an ilustrative example?
Comment: @Uri I would particularly like to see an example with Property (T).
Comment: This desire will not be fulfilled, by a theorem of Hjorth. I will write an answer when I will have the time.
Comment: @Uri What if I just want two non-conjugate free ergodic actions that are orbit equivalent?  Or infinitely many?
Answer: We consider infinite countable groups and ask about the this property: admitting uncountably many mutually non-isomorphis free ergodic probability measure preserving (pmp) actions that are all mutually orbit equivalent.
You ask which groups satisfy this property. I do not know exactly. I think no one knows. But it is a big class of groups, maybe all groups which do not satisfy property (T).
In Corollary 2.6 of A converse to Dye's theorem Hjorth proves that property (T) groups do not satisfy this property.
On the other hand, amenable groups do satisfy this property. Indeed, any amenable group has uncountably many mutually non-isomorphic pmp actions (say, by entropy theory), but these are all orbit equivalent, by the celebrated Ornstein-Weiss theorem.
To see that there are many non-amenable groups satisfying this property, fix a pmp action of a non-amenable group $\Gamma_1$ on $X$ and uncountably many pmp actions of an amenable group $\Gamma_2$ on $Y_\alpha$. Then the non-amenable group $\Gamma=\Gamma_1\times \Gamma_2$ will satisfy this property, as witness by its actions on $X\times Y_\alpha$.
If you want to study this property further, I suggest you to start with the following two things: try to show that if a group has a quotient that satisfies this property then the group satisfies this property, and try to show that this property is a measure equivalence invariant.


Can I upgrade my MacBook Pro which has Leopard Mac OS X 10.5.8 to Snow Leopard 10.6.2?

Question: I have a MacBook Pro which has Leopard Mac OS X 10.5.8 and i want to upgrade this to Snow Leopard 10.6.2. Is it possible? If yes then give me some guidelines.
Comment: "iphone" is not an appropriate tag for this question.
Comment: The one caveat I would add is "Don't pay too much"! ;-) I'm not sure what the best price currently is, but you can get a retail, 1 system 10.6.3 install DVD from http://eshop.macsales.com/item/Apple/MC573ZA/ for around $31 when shipped USPS. I suggest trying to get a version of the install DVD with includes as many "updates" as possible. It may make your install smoother and will save time by not having to download as large an update later.
Answer: This really isn't the right forum for a question of this sort our sister site superuser.com would be. That said this is very easy; just buy 10.6 from Apple, backup your system, boot from the disk, follow the installation instructions then update your system once it's installed - it's that easy.
Answer: According to Apple here are the required specs.

Mac computer with an Intel processor
1GB of memory
5GB of available disk space DVD drive for installation
QuickTime H.264 hardware acceleration requires a Mac with an NVIDIA GeForce 9400M, GeForce 320M, or GeForce GT 330M graphics processor.
Developer tools require 1GB of memory and an additional 3GB of available disk space.
OpenCL requires one of the following graphics cards or graphics processors: NVIDIA GeForce 320M, GeForce GT 330M, GeForce 9400M, GeForce 9600M GT, GeForce 8600M GT, GeForce GT 120, GeForce GT 130, GeForce GTX 285, GeForce 8800 GT, GeForce 8800 GS, Quadro FX 4800, Quadro FX5600, ATI Radeon HD 4670, ATI Radeon HD 4850, Radeon HD 4870
64-bit support requires a Mac with a 64-bit processor.
Grand Central Dispatch requires a Mac with a multicore processor.


Read a property of a array with Mongoose

Question: This is the Schema:
var userschema = new mongoose.Schema({

   user: String,
   pass: String,
   imagen: [{

              title: String,
              author: String,
              description: String,
              index: Number,
              path: String,

           }]

});

I'm trying with 'foomodel.find', to find inside the imagen array of the Schema, a element with an specific property, I mean:
usermodel.find({ user: foo, 'imagen.index': 1 }, function (err, imagen){

        if(err) throw err;

        console.log(imagen);

});

In the console, I received this [], when I want to received the properties of the imagen array's element, that has the index as one. I checked it exist. Any solution...?
Thank's advance!
Answer: I think you need to use $elementMatch:
usermodel.find({ user: foo, "imagen": {"$elemMatch":{"index": 1 }},
    function (err, imagen){
        if(err) throw err;
        console.log(imagen);
});


What is the word beginning "hom..." meaning an explanation in terms of persons and their actions?

Question: The word just escapes me, but it is frequently used politically to describe a narrative which relies on explaining a complex matter in terms of the people involved in the issue.
Comment: @Josh I've just remembered - the expression I'm thinking of is *ad hominem*. My apologies for troubling you. Others should consider the question closed. Thanks.
Comment: Sounds like humanized. >"The humanized brand will communicate the message, either directly or in a subtle way, that the bank understands its customers as humans, cares about what they are feeling and is committed to meeting their needs."  thefreelibrary.com/Humanizing+bank+brands-a0374971457
Comment: Add a sample sentence please. It may help to find the word.
Answer: It's hard to know for sure without more context, but you may be thinking of ad hominem.
https://en.wikipedia.org/wiki/Ad_hominem
In politics (and logic/rhetoric more broadly), it refers to arguments that target a person or their character instead of targeting the merits of their argument.
It's considered a logical fallacy and is often used in political discourse.


What grass type is this - Sydney, Australia?

Question: Looking to oversow some patches of my lawn and am hoping to match up to the existing grass type. We are in Sydney, Australia. Does anyone know what grass type I have?
Answer: Look like a Bermuda Grass Cynodon dactylon to me. Very Invasive.
Wikipedia/Bermuda-grass

From Wikipedia:
Cynodon dactylon, commonly known as Bermuda grass, is a grass found
worldwide. It is native to Europe, Africa, Australia and much of Asia.
It has been introduced to the Americas.[2][3] Contrary to its common
name, it is not native to Bermuda and is in fact an abundant invasive
species there. In Bermuda it has been known as "crab grass" (also a
name for Digitaria sanguinalis).

Weed Alert/ Bermuda grass


Is there an order of difficulty for MSA?

Question: Has an order of grammatical difficulty for Modern Standard Arabic been discovered (or proposed) similar to that found for L2 learners of English?
Comment: Can you explain what you mean in more detail?  A bit more information about "an order of grammatical difficulty" might help people understand your question.
Comment: Some research has identified an order of grammatical difficulty for L2 learners of English: learners appear to learn grammatical constructs in a relatively invariable order, regardless of which language they speak as L1.  The differences between L1 and English have a greater influence on the time it takes to proceed through the order, rather than change the order itself.
Answer: I think you're referring to Order of acquisition - the idea that there is a fixed order in which learners acquire different aspects of grammar.
There's one article in this journal; an unedited version can be found here.
Answer: i think there is not, each Arabic country have it's own program of learning, most likely that this programs gonna be close to each others but there is no unified program of progression nor an unified certificate. i am not 100% certain, so i might be wrong.


Create a list for Attendance tracking and a column that determines if the logged time is late or not

Question: I want to create a simple SharePoint list with a form that only asks for three things: Team the person belongs to, Choose between LOGIN or LOGOUT, and Shift Schedule from a dropdown menu. Each Shift Schedule has a fixed login time. For example 7:00am. So I use the Created field as it already captures the time the person created the item for LOGIN or LOGOUT. Lets say the person creates the item at 1:05pm. Which means he/she is late 5 minutes. So I want to add a new column for "Status" which displays "Late" or "On Time". What should be my formula for the "Status" column to calculate correctly if the person is late or not?
Answer: You can use the following formula
<code>=IF([LOGIN/LOGOUT]="LOGIN",IF(VALUE(TEXT([Created],"hhmmss"))>70000,"Late","On Time"),"")
</code>
Here is my testing results:

(As I need to get the testing results, I use "date" time instead of "Created" time.)


Would BMP7 introduced to a Petri dish with cartilage cause chondrocytes to replicate?

Question: Chondrocytes are the cells that make up cartilage. As I understand it, the best growth factor that signals replication of chondrocytes is Bone morphogenetic protein 7.
If supplied with a sufficient source of nutrients and a growth factor, would those chondrocytes begin to replicate?
If you prepare a Petri dish full of blood and cartilage and add BMP7, would the cartilage start to grow? Are those two questions the same?
Answer: Hello and welcome Delaney Fitzpatrick.
BMP7 - Bone morphogenetic protein 7 - Plays a role in cartilage homeostasis.
Chondrocytes - A cell which has secreted the matrix of cartilage and become embedded in it.
Chondrocytes therefore make cartilage. Cartilage unfortunately has a poor regeneration capacity.
BMP7 stimulates Chondrocyte division
Naturally the question then becomes, does BMP7 also increase cartilage production?
A paper from 2009 - Chondrocytes that expressed increased levels of BMP7 in defected rabbit knees showed statistically significant repair when compared to controls.
Although an early study, it does suggest that BMP7 may improve cartilage repair.


Why don't we just use the outer measure?

Question: Why do we care so much to restrict to a $\sigma$-algebra of sets on which the outer measure is countably additive? When is countable sub-additivity not enough? Does it pose a problem when defining the Lebesgue integral?
Comment: Countable additivity is perhaps the most important property of a measure.  You can show that the outer measure is additive when any two sets are separated by positive distance, but we desire additivity beyond that.
Comment: I'm starting a bounty because I think this is a good question. Specifically, I think the question is, "what goes wrong if you try to define integration against an outer measure rather than a measure?" If nothing goes wrong, then why not formulate measure theory in terms of outer measures and forego all the complicatedness of measurability?
Comment: @N.Virgo See the construction of Lebesgue integral in Royden's Real Analysis (Proposition 3 on pp.79 3e for me)
Answer: Below, $\mu^*$ denotes Lebesgue outer measure and $\mu$ denotes its restriction to the measurable sets.

Remember how measurability enters the definition of the Lebesgue integral of a (for simplicity) nonnegative function $f$ on $[0,1]$: we look at functions $h$ which have $h(x)\le f(x)$ for all $x\in[0,1]$, take on only finitely many values, and have measurable preimages of each particular value. Each such $h$ has a corresponding "naive integral" $$\theta_h:=\sum_{y\in im(h)}y\cdot\mu\{x\in [0,1]: h(x)=y\},$$ and we define $\int_{[0,1]}fd\mu$ to be the supremum of these $\theta_h$s.
If we drop that last condition, we get garbage: for example, let $f(x)=1$ everywhere, let $(V_i)_{i\in\mathbb{N}}$ be a decomposition of $[0,1]$ into Vitali sets of equal outer measure $\epsilon>0$, and for each $n$ let $$g_n(x)={1\over \min\{n, v(x)\}}$$ where $v(x)$ is the unique $i$ such that $x\in V_i$. (Note that I'm assuming $0\not\in\mathbb{N}$ here for simplicity; if, like me usually!, you prefer $0$ to be a natural number, throw in a "$+1$" appropriately.)
On the one hand we always have $g_n(x)\le f(x)$ for all $n$ and $x$. On the other hand the "naive integral" of each $g_n$ is $\ge \sum_{i=1}^n{\epsilon\over n}$, which is a big problem since the harmonic series diverges! Basically, if we were to ignore measurability concerns in the definition of the Lebesgue integral, every positive function would wind up having infinite integral over $[0,1]$.
Comment: @CharlesHudgins Well, it's consistent with $\mathsf{ZF + DC}$ (= set theory with a weak form of the axiom of choice) that every set is measurable, so that's unavoidable. (This is a theorem of [Solovay](https://en.wikipedia.org/wiki/Solovay_model).)
Comment: This makes sense, though it is interesting to me that the counterexample is *highly* non-constructive.
Comment: Thank you for the comment. I didn't know a result like that existed. I knew that non-measurable sets were non-constructive, but didn't know they were consistent with a weakened form of choice.
Comment: *their nonexistence is consistent with a weakened form of choice.
Comment: @Noah Schweber: After careful and multiple reads, I understand a good amount of your answer ( that  is pretty shocking in itself ). One question I do have is why you have to define $g_{n}(x)$ as the $\frac{1} {(min(n, \nu(x)})$ rather than just $\frac{ 1} {\nu(x)}$. Thanks.
Comment: @markleeds The function $1\over \nu(x)$ takes on infinitely many values, so isn't one of the functions considered in the definition of the Lebesgue integral.
Comment: So, by using  the $min(n, \nu(x))$ you fixed that by making it countable ? Thanks.
Comment: @Noah Schweber: I probably did not word it correctly in my last comment but my question is: "How did putting the min $(n, \nu(x))$ in the denominator make the function not take on infinitely many values. I'm not clear on how that works. Thanks so much and I won't bother you about this again.
Answer: Instead of viewing the domain of the outer measure as being restricted from $P(\mathbb{R})$ down to $\mathcal{B}$ in order to achieve countable additivity we could view it from the opposite direction, by observing that the outer measure is naturally countably additive already on quite a large class of sets - namely the `system of intervals' $\mathcal{I} = \{$ all countable unions of intervals in $\mathbb{R}\ \}$ - a proof of this is given from the definition of outer measure in [1]. However we find that the class of sets $\mathcal{I}$ lacks a crucial property required for the domain of the outer measure - namely closure under set complementation (a counterexample is provided in the Appendix of [1]), which then motivates the idea of a $\sigma$-algebra and the Borel algebra $\mathcal{B}$. We extend the domain $\mathcal{I}$ to $\mathcal{B}$ to achieve the closure under set complementation and then find that the countable additivity property is preserved (eg see [2, Chap 2]).
However a further reason for requiring countable additivity is that it allows certain useful theorems on outer measure to be proved - as Axler [2, §2C, p41] says :

The countable additivity that forms the key part of the definition [of a measure] above allows us to prove good limit theorems.

To see how these 'good limit theorems' come about let us first go back to some basic intuitive ideas about what we would expect from a notion of measure on $\mathbb{R}$ :

We require some function $\mu : \mathcal{C} \rightarrow [0, \infty]$ defined on some class of sets $\mathcal{C}$ in $\mathbb{R}$ which contains all intervals, and for which $\mu(I) = l(I)\ \forall$ intervals $I$, ie $\mu$ extends the notion of length of an interval in $\mathbb{R}$ to some larger class of sets.
We require $\mu(D \cup E) = \mu(D) + \mu(E)$ when $D, E$ are disjoint and more generally finite additivity ie. $\mu(E_1 \cup \cdots \cup E_n) = \mu(E_1) + \cdots + \mu(E_n)$ when $E_1, \ldots, E_n$ are disjoint.
We require order preservation : ie. $D, E \in \mathcal{I}$ and $D \subseteq E \Rightarrow \mu(D) \leq \mu(E)$ (follows from 2)
$\mu(\emptyset) = 0$ (follows from 2)

Countable additivity and closure of $\mathcal{C}$ under set complementation are not properties we would initially expect of a measure. However as we see below both these properties are needed to prove certain desired theorems involving limits of measures.
Firstly, from the above properties we have the following limiting processes :
If $E_1 \subseteq E_2 \subseteq \cdots$ is an increasing sequence of sets in $\mathcal{C}$ then :
\begin{equation}
\lim_{n \rightarrow \infty} \mu(\bigcup_{k=1}^{n} E_k ) = \lim_{n \rightarrow \infty} \mu(E_n) = \sup \{ \mu(E_n) \} \in [0, \infty] \label{eq:limit-union} \tag{1}
\end{equation}
and, if $E_1 \supseteq E_2 \supseteq \cdots$ is a decreasing sequence of sets in $\mathcal{C}$ then :
\begin{equation}
\lim_{n \rightarrow \infty} \mu(\bigcap_{k=1}^{n} E_k ) = \lim_{n \rightarrow \infty} \mu(E_n) = \inf \{ \mu(E_n) \} \in [0, \infty] \label{eq:limit-intersection} \tag{2}
\end{equation}
It is then natural to ask the question whether in (\ref{eq:limit-union}) and (\ref{eq:limit-intersection}) it could make sense for the limit operation to be 'passed through' the function $\mu$ to give the following :
If $E_1 \subseteq E_2 \subseteq \cdots$ is an increasing sequence of sets in $\mathcal{C}$ then :
\begin{equation}
\bigcup_{n=1}^{\infty} E_n \in \mathcal{C}\ \hspace{1em} \mbox{and} \hspace{1em} \mu(\bigcup_{n=1}^{\infty} E_n) = \lim_{n \rightarrow \infty} \mu(E_n) \label{eq:limit-union-pass-through} \tag{3}
\end{equation}
and, if $E_1 \supseteq E_2 \supseteq \cdots$ is a decreasing sequence of sets in $\mathcal{C}$ then :
\begin{equation}
\bigcap_{n=1}^{\infty} E_n \in \mathcal{C}\ \hspace{1em} \mbox{and} \hspace{1em} \mu(\bigcap_{n=1}^{\infty} E_n) = \lim_{n \rightarrow \infty} \mu(E_n) \label{eq:limit-intersection-pass-through} \tag{4}
\end{equation}
Note that since any union $\bigcup_{n=1}^{\infty} E_n$ can be expressed as an increasing union $\bigcup_{n=1}^{\infty} (E_1 \cup \cdots \cup E_n)$, (\ref{eq:limit-union-pass-through}) requires that $\mathcal{C}$ must be closed under all countable unions (and hence contain all countable unions of intervals). And since any intersection $\bigcap_{n=1}^{\infty} E_n$ can be expressed as a decreasing intersection $\bigcap_{n=1}^{\infty} (E_1 \cap \cdots \cap E_n)$, (\ref{eq:limit-intersection-pass-through}) requires that $\mathcal{C}$ must be closed under all countable intersections (and hence contain all countable intersections of intervals).
However in addition to the above heuristic to motivate the limit properties (\ref{eq:limit-union-pass-through}) and (\ref{eq:limit-intersection-pass-through}), a concrete example where properties (\ref{eq:limit-union-pass-through}) & (\ref{eq:limit-intersection-pass-through}) solve a problem in real analysis outside of measure theory can be found in the Bounded Convergence Theorem for Riemann Integrals in Bartle [3, §22.14, p288], where the author has omitted a portion of the proof which can be supplied by properties (\ref{eq:limit-union-pass-through}) & (\ref{eq:limit-intersection-pass-through}). In Bartle & Sherbert [4, §7.2.5, p304], this BCT is also stated but the proof is omitted entirely, being described as 'quite delicate'. What is required is to show that if $\delta > 0$ and if $(E_n)$ is a sequence of sets in $[0, 1]$ with each $E_n$ containing a finite number of non-overlapping closed intervals with a total length $\geq \delta$, then there exists a point belonging to infinitely many of the $E_n$. This can be proved using (\ref{eq:limit-union-pass-through}) & (\ref{eq:limit-intersection-pass-through}) (together with the finite additivity, order preservation, and extension of interval length) as follows : we require to show that the set $E = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} E_m$ is non-empty, so letting $F_n$ denote the union of the above intervals within $E_n$ (so $F_n \in \mathcal{C}$ due to (\ref{eq:limit-union-pass-through})), and letting $F = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} F_m \subseteq E$, then we have from (\ref{eq:limit-intersection-pass-through}), $\mu(F) = \lim_{n \rightarrow \infty} \mu(\bigcup_{m=n}^{\infty} F_m)$, where $\forall\ n,\ \mu(\bigcup_{m=n}^{\infty} F_m) \geq \mu(F_n) \geq \delta$ (noting $F_n$ equals a finite union of disjoint intervals with a total length $\geq \delta$), so that $\mu(F) > 0$, and hence $F \neq \emptyset$ and thus $E \neq \emptyset$. In Bartle [3] the Bounded Convergence Theorem for Riemann integrals is subsequently used to prove the Dominated Convergence Theorem for Riemann integrals [3, §25.21, p359], which in turn is used to prove Stirling's Formula in [5]. Thus we see a number of important concrete applications of properties (\ref{eq:limit-union-pass-through}) & (\ref{eq:limit-intersection-pass-through}).
Consider now how we can prove the properties (\ref{eq:limit-union-pass-through}) and (\ref{eq:limit-intersection-pass-through}). Firstly note (\ref{eq:limit-union-pass-through}) must hold if any $\mu(E_n) = \infty$, by order preservation, so for (\ref{eq:limit-union-pass-through}) assume every $\mu(E_n) < \infty$. Since the $E_n$ are increasing, the sets $E_n \setminus E_{n-1}$ are disjoint. Thus provided each $E_n \setminus E_{n-1} \in \mathcal{C}$ and $\mathcal{C}$ is closed under countable union we would obtain :
\begin{eqnarray}
\mu(\bigcup_{n=1}^{\infty} E_n) & = & \mu(\bigcup_{n=1}^{\infty} (E_n \setminus E_{n-1}) ), \hspace{2em} \mbox{defining } E_0 = \emptyset \nonumber \\
& = & \sum_{n=1}^{\infty} \mu(E_n \setminus E_{n-1}), \hspace{2em} \mbox{if countable additivity applied} \nonumber \\
& = & \lim_{n \rightarrow \infty} \sum_{k=1}^{n} \mu(E_k \setminus E_{k-1}) \label{eq:telescope} \tag{5}
\end{eqnarray}
But again applying countable additivity we would have within $\overline{\mathbb{R}}$ :
$$\mu(E_k) = \mu(E_{k-1}) + \mu(E_k \setminus E_{k-1})$$
and so as every $\mu(E_k)$ is in $\mathbb{R}$ :
$$\mu(E_k \setminus E_{k-1}) = \mu(E_k) - \mu(E_{k-1})$$
so that (\ref{eq:telescope}) telescopes to give the required (\ref{eq:limit-union-pass-through}). The countable additivity was essential at two critical points of the argument. We note the identity $\mu(E \setminus D) = \mu(E) - \mu(D)$ holds whenever $\mu(D) < \infty$.
In the above, as well as the countable additivity of $\mu$ and the closure of $\mathcal{C}$ under countable union we required that the set difference $E_n \setminus E_{n-1} \in \mathcal{C}$ for any $E_n$ and $E_{n-1}$ in $\mathcal{C}$. Since $\mathcal{C}$ contains all intervals, and hence $\mathbb{R}$ itself, we thus require in particular that $\mathbb{R} \setminus E \in \mathcal{C}, \forall\ E \in \mathcal{C}$, ie. $\mathcal{C}$ is closed under set complementation. From De Morgan's laws the latter together with closure under countable union implies closure under countable intersection.
Property (\ref{eq:limit-intersection-pass-through}) readily follows on from (\ref{eq:limit-union-pass-through}), with no additional properties of the measure being required, but the condition $\mu(E_1) < \infty$ is required :
Since $(E_n)$ is decreasing $(E_1 \setminus E_n)$ is increasing and so applying (\ref{eq:limit-union-pass-through}) we have :
\begin{eqnarray*}
\mu(\bigcup_{n=1}^{\infty} E_1 \setminus E_n) & = & \lim_{n \rightarrow \infty} \mu(E_1 \setminus E_n) \\
\mbox{ie.} \hspace{3em} \mu(E_1 \setminus \bigcap_{n=1}^{\infty} E_n) & = & \lim_{n \rightarrow \infty} \mu(E_1 \setminus E_n), \hspace{2em} \mbox{by De Morgan.}
\end{eqnarray*}
But
$$\mu(E_1 \setminus \bigcap_{n=1}^{\infty} E_n) = \mu(E_1) - \mu(\bigcap_{n=1}^{\infty} E_n)$$
and
$$\mu(E_1 \setminus E_n) = \mu(E_1) - \mu(E_n)$$
so
$$\mu(E_1) - \mu(\bigcap_{n=1}^{\infty} E_n) = \lim_{n \rightarrow \infty} (\mu(E_1) - \mu(E_n)) = \mu(E_1) - \lim_{n \rightarrow \infty} \mu(E_n)$$
therefore, since $\mu(E_1) < \infty$, (\ref{eq:limit-intersection-pass-through}) follows.
Thus in summary, without the countable additivity property we would not be able to obtain the required limit theorems (\ref{eq:limit-union-pass-through}) and (\ref{eq:limit-intersection-pass-through}), and we can see the necessity of this property prior to considering the Lebesgue integral. In summary the properties required of $\mu : \mathcal{C}\ \rightarrow [0, \infty]$ for it to satisfy both the above intuitive notions of a measure and the desired limit theorems are :
(A) For the domain $\mathcal{C}$ of $\mu$ :

contains all intervals
is closed under countable union
is closed under set complementation

(B) For the function $\mu$ :

satisfies $\mu(I) = l(I)\ \forall$ intervals $I \subseteq \mathbb{R}$
is countably additive, ie $\mu( \bigcup_{n=1}^{\infty} E_n ) = \sum_{n=1}^{\infty} \mu(E_n)$, whenever $(E_n)$ are disjoint sets in $\mathcal{C}$

In Axler [2] proof that countable subadditivity follows on from these properties is given in Theorem 2.58, and the above limit theorems are given as Theorems 2.59 and 2.60.
References
[1] Ross Ure Anderson (2023), Intervals and Outer Measure on $\mathbb{R}$, https://ia801904.us.archive.org/19/items/intervals-and-outer-measure-on-r/IntervalsAndOuterMeasureOnR.pdf
[2] Sheldon Axler (2020), Measure, Integration & Real Analysis, Springer Graduate Texts in Mathematics, https://measure.axler.net/.
[3] Robert G. Bartle (1964), The Elements of Real Analysis, John Wiley & Sons.
[4] Robert G. Bartle & Donald R. Sherbert (1982), Introduction to Real Analysis, John Wiley & Sons.
[5] Keith Conrad, Stirling's Formula, https://kconrad.math.uconn.edu/blurbs/analysis/stirling.pdf


How to design deep convolutional neural networks?

Question: As I understand it, all CNNs are quite similar. They all have a convolutional layers followed by pooling and relu layers. Some have specialised layers like FlowNet and Segnet. My doubt is how should we decide how many layers to use and how do we set the kernel size for each layer in the network. I have searched for an answer to this question but I couldn't find a concrete answer. Is the network designed using trial and error or are some specific rules that I am not aware of? If you could please clarify this, I would be very grateful to you.
Answer: Short answer: if there are design rules, we haven't discovered them yet.
Note that there are comparable questions in computing.  For instance, note that there is only a handful of basic electronic logic units, the gates that drive your manufacturing technology.  All computing devices use the same Boolean logic; some have specialised additions, such as photoelectric input or mechanical output.
How do you decide how to design your computing device?

The design depends on the purpose of the CNN.  Input characteristics, accuracy, training speed, scoring speed, adaptation, computing resources, ... all of these affect the design.  There is no generalized solution, even for a given problem (yet).
For instance, consider the ImageNet classification problem.  Note the structural differences between the winners and contenders so far: AlexNet, GoogleNet, ResNet, VGG, etc.  If you change inputs (say, to MNIST), then these are overkill.  If you change the paradigm, they may be useless.  GoogleNet may be a prince of image processing, but it's horrid for translating spoken French to written English.  If you want to track a hockey puck in real time on your video screen, forget these implementations entirely.

So far, we're doing this the empirical way: a lot of people try a lot of different things to see what works.  We get feelings for what will improve accuracy, or training time, or whatever factor we want to tune.  We find what works well with total CPU time, or what we can do in parallel.  We change algorithms to take advantage of vector math in lengths that are powers of 2.  We change problems slightly and see how the learning adapts elsewhere.  We change domains (say, image processing to written text), and start all over -- but with a vague feeling of what might tune a particular bottleneck, once we get down to considering certain types of layers.
Remember, CNNs really haven't been popular for that long, barely 6 years.  For the most part, we're still trying to learn what the important questions might be.  Welcome to the research team.
Comment: I also have the same question. Although, your opinion is accurate in concluding that an empirical approach is chosen, I am not able to understand if there is at least some design process to start making a model or not? It will be too random to just put layers on top of one another and expect them to give a somewhat accurate result. There should be some basic guidelines on how to start and then empirical methods can be used to fine tune the model.
Comment: @Qazi At the level you're asking the question, there are no such guidelines.  It sounds as if you're asking for practical, applicable guidelines for "how do I make a model?"  Until you classify the model according to it's general characteristics, we can't even say that a CNN might be a good solution.  For most modelling situations, a neural network is a waste of computing resources.
Comment: @Qazi You are correct that it's not practical to just start slapping layers together.  Rather, you need to analyze your input texture, consider your desired modelling purpose and performance, determine what features you could derive from the input that *may* lead to the output you want, and then experiment with network topologies that embody those features.  A relatively small change in the input texture often results in a large change in the model topology.
Comment: Thank you for your complete answer. However, it is almost 2 years later. Is an update worth mentioning? I have the same question, also for RNN and LSTM.
Comment: Yes, an update is *absolutely* worth mentioning!
If you have something to add, please do.  You prompted me to add a link I found around the turn of the year.  Your question on RNN and LSTM is perfectly valid; please post as a separate question, linking to this one for reference.
Comment: Thanks for the complete answer, it was very helpful. Does somebody know an alternative URL? The link mentioned was changed by Kaggle.


Should I say 'in 3D' or 'on 3D'?

Question: Should I say "I've watched the film in 3D" or "I've watched the film on 3D"?
Answer: The former ( in 3D)
Here's a straightforward  example from Longman :"a film in 3-D"
Answer: 3D means three dimensions. Something cannot be seen on those dimensions, but can be seen in them. Had the phrase been '3D t.v.', then it could be seen on a 3D t.v. No mention of that in the question though...

Should I say 'in 3D' or 'on 3D'?


Is it possible to make consoles full screen?

Question: Is it possible on to switch a console to full-screen in Windows XP on a single monitor? Meaning on a dual monitor that only the left (or right) screen displays the console while the other one remains on Windows. By full-screen I mean true-fullscreen mode, not just resizing (maximize) to fill in the screen.
Comment: Windows Vista/7 or XP?
Comment: Windows XP (no choice in office)
Answer: Windows 7 (and I believe Vista) doesn't support the full-screen feature for MS-DOS or Command Prompt console. It doesn't even allow full-screen when using the Command Prompt via Windows 7's Windows XP Mode. On Windows XP, it can be set to full screen via Alt + Enter or right-clicking the console title bar and selecting Full Screen.
A possible workaround on Windows 7: With Command Prompt open, right click the console title bar, then select Properties. Within the Layout tab, you can adjust the width and height. My suggestion is: Adjust only the width to match the width of your screen. Once the adjustments are made, double click the console title bar. Using this workaround gives it the closest full-screen effect that can be achieved.
Comment: `On Windows XP, it can be set to full screen via Alt + Enter or right-clicking the console title bar and selecting Full Screen.`   It might not work without the right driver.
Comment: This is correct.  The newer display drivers no longer suppport full screen console programs.  Now if you switch back to XP drivers. ..
Answer:
Is it possible to make consoles full screen?

Why yes; yes it is, but only up to Windows XP, it is no longer supported in Vista and up. You can get basic full-screen functionality with just Alt+Enter, but for better compatibility and to resolve some issues, you can install the Full screen video driver for console:

Open the Add Hardware applet in the Control Panel
Click Next, then select Already Connected
Select Add a new hardware device at the bottom of the list
Choose Manually Select (Advanced)
Select Show All Devices
Under Manufacturer, select Microsoft Corporation
Under Model, select Full screen video driver for console
Next, Next, Finish (reboot if prompted)

Now you can press Alt+Enter in console windows to switch them to fullscreen mode (it displays in whichever monitor is set to the primary).

NB: this works best for the command-prompt (ie, text-mode) while graphics-mode games and such tend to have problems (eg hanging). If you want to get games working in full-screen (or at all), then you need to use various hacks to get them to work such as using the nolfb[1] and cli2nop[2] tools and VDMSound.
Comment: run `devmgmt.msc` menu `action`, `add legacy hadrdware` for me in windows7 ***x64***. But ***I have no "Full screen video driver" there***. Also I have ***no `fsvga.inf` on my system drive C:***
Comment: @naxa, because it is [no longer supported in Windows Vista](http://superuser.com/questions/314675/what-are-the-requirements-for-fullscreen-consoles-in-windows-7) and up; it only works up to Windows XP (which is what this question is about).
Comment: ah. there was some hackery where you could supposedly turn full screen on 7 back on, most likely due to a windows bug but I cannot find it.
Comment: @naxa, the closest you can get is to simply maximize the console window; there's no way (that I've been able to find in my quest) to make it actually fullscreen in 7 like you could in XP (where the screen actually changes to text-mode). ◔̯◔
Answer: ShiftWindow : Resize an app window to get pseudo fullscreen, this will hide titlebars and window borders for msot applications, you may need to actually increase cmd.exe window width from the properties dialog first to be effective.
I haven't tested the above but this might work, good luck :)
Comment: How is DualMon actually related to having a console full-screen on only one monitor? ShiftWindow may kind of resize to fill in the screen but that seem to be for EVE only.
Comment: @Wernight ShiftWindow will work for any application, you just need to define a Rule with proper window title.
Answer: Cygwin has a custom terminal that will go "true" full screen.
By default only the scrollbar on the right will be visible. You can turn those off in options if you desire, and also enable block cursor or whatever other flavor you like. It's a completely custom Terminal implementation, not a regular windows window console app.
To toggle cygwin terminal fullscreen mode simply press Alt+Enter
Incidentally the cygwin terminal also supports transparency, if you like to have your log or output in a separate terminal underneath your working terminal, like you can on most linux distros.


What kind of cabbage to use in cabbage poriyal (Indian stir-fry)?

Question: I tried making this with a savoy cabbage, and it tastes like Irish cabbage - the spices were overwhelmed by the cabbage taste. When I have had this dish in Indian restaurants, the cabbage taste is milder.
Comment: I've never had authentic cabbage poriyal, hence this is a comment and not an answer, but I'd try [napa cabbage](http://en.wikipedia.org/wiki/Napa_cabbage).  It's relatively mild in flavor and very common in East-Asian cuisine.
Answer: From what I can find, it seems like the cabbage used for many Indian dishes is what we in the US would call green cabbage. In the pictures of cabbage poriyal that I saw, the cabbage appeared to have a smooth texture rather that a crinkly texture like Napa or Savoy cabbage would have.
Here are a couple of excerpts from  an article about cabbage use in Indian cooking :

Cabbage commonly known as Patta gobhi or Band gobhi in India is a very integral part of Indian cooking. It is usually used to prepare salad, soups, pickles, curries, kootus, added to dals and cooked with other various vegetables etc. There are many types of cabbage available in market which is different in colour, their flavours, texture etc.
There are many varieties of cabbage based on shape and time of maturity. Cabbage has a round shape and is composed of superimposed leaf layers. There are three major types of cabbage: green, red, and Savoy. The color of green cabbage ranges from pale to dark green. Both green and red cabbages have smooth-textured leaves. Red cabbage has leaves that are either crimson or purple with white veins running through it. The leaves of Savoy cabbage are more ruffled and yellowish-green in color. Red and green cabbages have a more defined taste and crunchy texture as compared to Savoy cabbage's more delicate nature.

Here is a picture of cabbage from a recipe for an Indian curry. It does appear to be a typical green cabbage.


Reduce Clique to Set Cover

Question: Is it possible directly to reduce clique to set cover?
I know that there are some ways of direct reduction from Clique to Vertex Cover and from Vertex Cover to Set Cover, so I am very interested to know if the is a way to reduce clique to set cover directly without the use of the transitive rule.
Comment: Welcome to CS.SE!  If you just compose those two reductions that *gives* you a direct reduction.  Have you tried working out what reduction mapping that yields, if you try that?  It seems like that immediately gives you what you want.
Comment: I tried it, but steel can't compouse direct flow, can you please suggest a way how it can be compoused.
Answer: Both problems are NP-complete and, by definition, there is a direct reduction between any pair of NP-complete problems.

Reduce Clique to Set Cover


Restoring the exact original situation of a spatial bookmark?

Question: After saving a view with a spatial bookmark, if one later wants to get back to this exact same view, it appears that you have to have the correct height/width ratio in the view panel. Is there a way to save and reestablish that?
Answer: This might be a way, though not automatic:
When you save a view (spatial bookmark) the way you want it (the horizontal field of view as well as vertical), note the scale.
Then when you wish to come back to this exact same view, first set the view panel so it is wider than the way you want it. Click on and "zoom to" the spatial bookmark. The scale will be correct but your horizontal field will be too large. Drag the left edge of the view to the right, until the map scale (shown in a box at the bottom) begins to change. That's when you have gone too far. Drag it back to the left just until your resolution is again correct. You then have the exact same view (I think...)
Comment: I'd recommend then that the "scale" be made a part of the bookmark name. It's really too bad that SpatialBookmarks don't record the enclosing window geometry because I use that geometry to "frame" the area I want to print in the print composer, and then save off a spatial bookmark of the view. More often than not, I need to add a new layer that I forgot and print it out too, and if the windows geometry has changed in the meantime due to other projects, my hope for the new layer registering with the existing ones is near-zero. I typ. have to reprint all layers. Thank you very much for the sugg.
Comment: If an interested person is listening, this would be a great plug-in: "Remember/Set my viewport."
Comment: I have tried this--and thank you again.  The idea is sound, but for the large areas my projects cover, and the small display I use, it is too coarse an approach. "Shirley," there must be programmatic solution!
Answer: I present my sorry, ugly, brutish attempt at recording, at least, the 'view' parameters for later user-assisted restoration.  And I have to say, it almost works.
I have QGIS set to open the Python console on startup (although I've forgotten how) and, accidentally, the script editor opens onto this, the only script I've ever written, or and possibly likely to write.
After I "save" a view with a Spatial Bookmark, I then run my script which:

saves a file QGIS_<project_name,etc.>.txt in the project folder,
containing all the relevant parameters for reconstructing the view.
lists, as a 'help', a set of commands to be copied/pasted to the
open Python console to re-create the view later. (The

In practice, after running the script, I open QGIS and the project at some later time. I open the script and, if I have not run the script on another project since, copy/paste the commands to re-create the view. (I believe it may help to have the QGIS window large but not maximized.) If otoh I have run the script on another project, I open the previously saved parameters file and copy paste the relevant parameters into the commands before executing them in the Python console. You can tell my "style" is in the vein of "monkey-see/-do."
<code># open Plugins->Python Console
# references for these functions...
#  https://qgis.org/pyqgis/3.2/gui/Map/QgsMapCanvas.html#qgis.gui.QgsMapCanvas.setScaleLocked
#  https://qgis.org/pyqgis/3.2/core/other/QgsRectangle.html?highlight=qgsrectangle
#  https://qgis.org/pyqgis/3.2/core/Point/QgsPointXY.html?highlight=qgspointxy
#  https://doc.qt.io/qtforpython/PySide2/QtCore/QSize.html
pn=QgsProject.instance().fileName()
pp=QgsProject.instance().homePath()
pn, pp
pfi=QFileInfo(QgsProject.instance().fileName())
fne=pfi.fileName()
fn=fne[:fne.find('.qgz')]
pfi.absolutePath()

canvas = qgis.utils.iface.mapCanvas() #create shortcut to the current canvas
ws=canvas.size()
sc=canvas.scale()
sl=canvas.scaleLocked() # T/F
mf=canvas.magnificationFactor()
ex=canvas.extent()
cp=canvas.center()
ws,sc,sl,cp
ex
# write these out to a file
#f=open('qgis_view_data.txt','w')
f=open(str(fn)+'_qgis_params_to_reconstruct_view.txt','w')
# NB write takes a single, string, argument
f.write('Project:\n'+str(pn)+'\n')
f.write('   File:'+str(fne)+'\n\n')
f.write('extra spatial bookmark data nec. to\n')
f.write(' reconstruct view in QGIS in order to\n')
f.write(' assure perfect registration for new\n')
f.write(' layers.\n')
f.write('generated by a crude pygis script of my device; namely,\n')
f.write('  ./upundn_projects/_NB_QGIS/save_qgis_view_parameters.py\n')
f.write('which outputs this local file:\n')
f.write('    <PROJECT>_qgis_params_to_reconstruct_view.txt\n')
f.write('containing the parameters nec. to recreate the view layout; e.g.,\n')
f.write('  WindowSize(px): QSize('+str(ws.width())+','+str(ws.height())+')\n')
f.write('     ActualScale: '+str(sc)+'\n')
f.write('ScaleLocked(T/F): '+str(sl)+'\n')
f.write('    MagnifFactor: '+str(mf)+'\n')
f.write('     Extents(km): QgsRectangle('+str(ex.xMinimum())+','+str(ex.yMinimum())+','+str(ex.xMaximum())+','+str(ex.yMaximum())+')\n')
f.write(' CenterPoint(km): QgsPointXY('+str(cp.x())+','+str(cp.y())+')\n\n')
f.write('Note:ActualScale=DisplayedScale*MagnifFactor\n\n')
f.write('Before running the script, and thus saving the parameters:\n')
f.write('  Create a new shapefile layer.\n')
f.write('  Select layer and toggle Edit mode. Use Azimuth Sequence Digitizer to\n')
f.write('  Draw a framing rectangle of the desired 3:2 aspect ratio.\n')
f.write('  Move it to the desired location.\n')
f.write('  Set a large 3:2 AR window size: canvas.resize(QSize(750,500).\n')
f.write('  [Note: Window Units do Not Align with Projection Units.]\n')
f.write('  Roughly zoom to that rectangle and note the scale and magnif.\n')
f.write('  [Note: scale400k/magn100 & scale500k/magn120 are the same view.]\n')
f.write('  Adjust window edges to closely fit the rectangle.\n')
f.write('  Create a spatial bookmark.\n')
f.write('Finally, run the script to save the parameters of the view.\n\n')
#
# presumably there is some way to use this information to
#  reconstruct the view with "set" type functions
f.write('To reconstruct view, reconstruct the conditions; e.g., in the console,\n')
f.write('copy/paste these commands):\n')
f.write('  canvas = qgis.utils.iface.mapCanvas() # attach to the iface\n')
f.write('  ws=QSize('+str(ws.width())+','+str(ws.height())+') # set window size\n')
f.write('  canvas.resize(ws)\n')
f.write('  sc='+str(sc)+' # set window scale factor\n')
f.write('  canvas.zoomScale(sc)\n')
f.write('  sl='+str(sl)+' # lock scale for zooming (also bool(1) but not true)\n')
f.write('  canvas.setScaleLocked(sl)\n')
f.write('  mf='+str(mf)+' # set magnification factor/zoom\n')
f.write('  canvas.setMagificationFactor(mf)\n')
f.write('  ex=QgsRectangle('+str(ex.xMinimum())+','+str(ex.yMinimum())+','+str(ex.xMaximum())+','+str(ex.yMaximum())+')\n')
f.write('  canvas.setExtent(ex) # set XY extents\n')
f.write('  cp=QgsPointXY('+str(cp.x())+','+str(cp.y())+') # set center point\n')
f.write('  canvas.setCenter(cp)\n')
f.write('Note: setting ws and ex should result in cp and sc being set as well.\n')
f.close()

print(fn+' view parameters saved')
</code>
I wrote "almost works" because it seems at times that some of the restore commands are ignored, and do not change the view. This is where a monkey s/d approach breaks down.


Can blood group affect behavior?

Question: In Japan, blood group is believed to have ties with behavior. Is there scientific evidence for it?
Comment: This claptrap about blood type and personality has also spread to Korea. :-(
Comment: If they believe it then it will almost probably be measurably so. But only in the fact that people often become what it is that they are expect to be, just via social expectation. Whether or not people with certain blood groups are statistically more likely to be born in certain areas which again impose a social responsibility on the bodies involved, is yet another factor. Genetically it may be possible that certain blood groups bring with them slightly different patterns in brain developments, however that is a big claim. --- For this to be true, (and not just a measurable effect of Japanese
Answer: A good overview about this including references is given by
Does blood type affect vocational behavior? by Andrew D. Carson, Ph.D.
In summary it states that an influence of the blood type on the behavior can not be fully dismissed out of hand, but there are also research results which did not found any connection.
Except: (emphasis mine)

There have been a number of studies over the years testing relationships between blood type and personality, and the following brief review is illustrative rather than definitive. I provide this review purely to demonstrate that claims of a relationship between blood type and personality should not be dismissed out of hand. Early published reports (Koga & Kato, 1934; Thompson, 1936) in general found no relation between blood type and personality traits or abilities.
However, more recent research has tended to find some linkages. Rinieris, Stefanis, and Rabavilis (1980), in a large (N=600) study of normal individuals, found greater incidence of obsessional personality traits in blood types A, B, and AB than in type O. In a study of 137 graduate students, Marutham and Prakash (1990) found Type B individuals to have higher neuroticism scores (compared to types O and A) on the Eysenck Personality Inventory; there were no differences on either extraversion or Type A behavior pattern. In a small study of individuals with duodenal ulcers, Neumann, Shoaf, Harvill, Jones, and Edward (1992) reported that Type A individuals had higher trait anger, trait anxiety, and depression compared to Type O individuals on standard measures of personality.
If there are indeed relations between blood type and personality, then it is not impossible that there may also exist differences in typical blood type across occupations, and that some blood types are associated with greater performance or resistance to occupational stress than are others within given occupations. In other words, it may be possible to apply a person-environment fit theory of work using blood type as a person-attribute. However, I recommend a more thorough literature review, followed by empirical research directly testing this notion before engaging in further speculation along these lines.

I also found an journal paper which claimed a (small?) relationship between blood type and personality:
Blood Groups and Personality Traits, by R. B. CATTELL, H. BOUTOURLINE YOUNG, AND J. D. HUNDLEBY, published in the AMERICAN JOURNAL OF HUMAAN GENETICS, VOL. 16, No. 4 (DECEMBER), 1984.

SUMMARY:
The High School Personality Questionnaire, Form A (10 items per factor)
was given to 581 Italian or Italo-American boys averaging 14 years of age in
four different cities. They were also classified for ABO types. Analysis of
variance was used to detect associations between ABO type and the 13 personality
factors and intelligence.
An association beyond the P < .01 level was found for I factor, premsia vs.
harria ("tender-minded vs. tough-minded"), A blood type being more premsic,
and 0, B, and AB more harric, in that order. A relationship which appeared
to warrant further investigation was personality factor J, coasthenia, with
blood group AB.
Answer: There is no scientific evidence that blood group affects one's behavior or personality.
The history of this pseudoscience is interesting. Paraphrased from the Blood types in Japanese culture Wikipedia entry:

Different blood types were first identified by Karl Landsteiner in 1900, and soon after it was noted that there were different blood group distributions around the world. Sadly, this information was used for racial prejudice.
The notion of blood group affecting behavior became popularized in Japan in 1927, when Takeji Furukawa, a professor at Tokyo Women's Teacher's School, published his paper "The Study of Temperament Through Blood Type" in the scholarly journal Psychological Research. The idea quickly took off with the Japanese public despite Furukawa's lack of credentials.
Interest in this pseudoscience started to wane in the 1930s, but was revived in Japan in the 1970s with a book by Masahiko Nomi, a lawyer and broadcaster with no medical background. Nomi's work was largely uncontrolled and anecdotal, and the methodology of his conclusions was unclear.

In a nutshell, one's blood group is about as telling of their temperament and personality as is their Zodiac sign. Which is to say, not at all.
Comment: Scott, I suggested an edit to your answer to correct two minor issues. I think a direct quotation may have better served this answer, as some sentences in the paraphrased section are very similar to the original.
Comment: As mentioned in most sources on "Blood type theory", it is worth noting that not only is the theory complete pseudo-science, but is also rooted in a tradition of early 20th century racially-motivated "research". 1930s Japan (when the theory took off) was in many ways similar to Germany of the same era: there was a strong push by authorities to justify their imperialist and racist policies through science.

Can blood group affect behavior?


Gaussian elimination with 4 variables

Question:
After this step however, I end up with 2 simultanious equations but still with four unknowns, I'm not sure if my gaussian elimination is wrong or not but how will i find the exact values of x,y,z and w?
Comment: Do you know about determinants?  What is the determinant of the original $4 \times 4$ matrix?  Do you know what it means to the system $Ax =b$ if $\det(A) = 0$?
Comment: It is the product of all the numbers diagonally from the top left to the bottom right?
Comment: No it's not the product of the diagonal.  The sum of the diagonal is called the matrix trace, but the [determinant](https://en.wikipedia.org/wiki/Determinant) is different altogether.  In your case since $\det(A) = 0$, then $Ax = b$ will not have a unique solution.
Answer: If the solution you reach is correct, then, $y$ and $w$ can take any value, and $x$ and $z$ are equal to $x=100-3y+96w$ and $z=54+52w$.
$y$ and $w$ can take any value because the equations 3 and 4 are equivalent to: $0x+0y+0z+0w=0$ and from here, because those equations are pivotal: $0z=0$ and $0w=0$.
As you see, because the gaussian elimination discarded 2 equations, we have 4 variables and 2 LI equations, thus the space of available solutions has dimension 4-2=2.
$y$ and $w$ are the 2 free variables parametrizing that solution space.
Actually, you can say that, i.e. $x$ and $w$ are your 2 free variables. The problem is the same in this case, if you reinterpret the problem by swapping the columns and proceeding as usually.

Gaussian elimination with 4 variables


GR correction to perihelion precession comparing with newtonian orbit

Question: My professor did the following derivation of the formula for the perihelion precession $\delta \phi = \frac{6\pi G M}{a(1-e^2)}$, but I am missing a factor of 2. I would appreciate if someone can help me spot the mistake, or tell me if the approach is totally wrong.
Consider the Schwarzschild metric, and compute the length of a geodesic as:
$$S = \int_0^{\tau} d\tau' \sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}} = \int_0^{\tau}d\tau' \sqrt{2T}.$$
Since the modulus of four-velocity is conserved and equal to $-1$, we have $2T=1$, and imposing the variation of the length equal to zero we get:
$$\delta S = \int_0^{\tau} d\tau' \delta\sqrt{2T} = \int_0^{\tau} d\tau' \frac{\delta T}{\sqrt{2T}} = \delta \int_0^{\tau} d\tau'
 T.$$
Therefore, we have reduced to the lagrangian formalism where the lagrangian is given by $T$ and its functional value is conserved and equal to $1/2$:
$$T = \frac{1}{2}\left[\left(1-\frac{2GM}{r}\right)\dot{t}^2 - \left(1-\frac{2GM}{r}\right)^{-1} \dot{r}^2 - r^2 \dot{\theta}^2 -r^2 \sin^2(\theta)\dot{\phi}^2 \right] = \frac{1}{2}$$
We can check using the Euler-Lagrange equation for $\theta$ that $\theta(\tau) =\frac{\pi}{2}$ is a solution, therefore the term $r^2\dot{\theta}^2$ disappears (since $\theta$ is constant) and the term $r^2 \sin^2(\theta) \dot{\phi}^2$ simplifies to $r^2\dot{\phi}^2$.
At this point we can use the Euler-Lagrange equation for $t$ and $\phi$ and see that there are two conserved quantities:
$$E = \frac{\partial T}{\partial \dot{t}} = \left(1-\frac{2GM}{r}\right)\dot{t}$$
which is the energy, and the angular momentum:
$$L = - \frac{\partial T}{\partial \dot{\phi}} = r^2 \dot{\phi}.$$
If we invert the two above relations for $\dot{t}$ and $\dot{\phi}$ and plug them into the expression for $T$, we get a first order differential equation for $r$. After some algebraic manipulations we end up with:
$$\frac{1}{2} \dot{r}^2 + \frac{L^2}{2r^2} + \frac{1}{2}\left(1-\frac{2GM}{r}\right) - \frac{GML^2}{r^3} = \frac{E^2}{2},$$
where we recognize the kinetic term $\frac{1}{2}\dot{r}^2$, so the rest of the left-hand side is an effective potential $V(r)$.
Now, we can look for newtonian circular orbits by neglecting the term $\sim 1/r^3$, setting the first derivative of the potential to $0$ and solving for $r$: we get
$$r_c = \frac{L^2}{GM}.$$
Now, we can expand the potential around the circular orbit, obtaining a harmonic potential, with the second derivative of the potential evaluated at $r=r_c$ playing the role of $\omega^2$, the square of the frequency of the harmonic oscillator. If we compute the second derivative at $r=r_c$ we get (I checked it with WolframAlpha):
$$V''(r_c) = \omega^2 = \frac{G^4M^4}{L^6}\left(1-\frac{12G^2M^2}{L^2}\right),$$
where my professor claims that $\omega_N^2 = \frac{G^4M^4}{L^6}$ is the Newtonian orbit frequency.
Now, since the GR orbit has a lower frequency than the newtonian one, it means that when the GR orbit has finished a revolution ($\phi = 2\pi$), the newtonian has travelled an additional angle given by:
$$\delta \phi = \omega_N \delta T = \omega_N \left(\frac{2\pi}{\omega} - \frac{2\pi}{\omega_N}\right) = 2\pi\left(\frac{1}{\sqrt{1-\frac{12G^2M^2}{L^2}}} - 1\right) \simeq \frac{12\pi G^2M^2}{L^2},$$
which clearly differs from the famous formula (with eccentricity equal to zero) just by a factor of $2$.
I double-checked the steps multiple time, but there not seem to be calculation errors. The only thing I can think of is that there is some wrong physical assumption somewhere, but I am not sure.
Answer: Orbital precession is found from the difference between the frequency in $\phi$ and the frequency of oscillation in $r$. Both calculated in the Schwarzschild metric and both of which differ from the Newtonian result, in opposite directions.


What crafted items have the best sustainable profit margins in Rage?

Question: I was wondering, does anyone know which recipes have the highest sustainable profit margins in Rage? Specifically, recipes with ingredients I can repeatedly purchase from merchants.
Comment: Sorry, this is way off topic, but: Is rage a good game or an awesome game? What would you compare it to? I'm wondering if I should get it...
Comment: Rage is like Half Life 2 meets Borderlands. The driving, aiming and weapons are all better but there's less to do and less to shoot at.
Answer: After playing through, the best answer I can come up with is that there really isn't a recipe good for your personal economy. Instead, craft lots of wingsticks and pop rockets and go do a few rounds of Mutant Bash TV. You can average something like $600-$700 per episode.
Answer: The best way to make money I've found is once you get to Subway Town, go to Stew, the man upstairs who sells one product at a time for cheap. Most of his products will make you lose money, but if he's selling the Viper Rockets, simply buy all you can and then go sell them to the shop a little ways away. Do this over a bunch of times and you'll reach the $10,000 in-game limit (lame!). I used this to max out all of my ammo, shields, homing rockets, you name it and left with a full $10,000 to boot.
If he's selling something else, just go outside to the Wasteland and then come right back into Subway Town and he should be selling something else. Do this until he's selling the Viper Rockets.


Accelerometer to track Potholes

Question: So I am trying to make an application to track potholes in a road. I am using accelerometer readings for that. Assuming that the phone is mounted in a car and our orientation doesn't change. After removing noise (through a low pass filter) I am saying that if I find a peak greater than a threshold in say the y axis I would like to classify that as a pothole. But I feel that this algorithm would confuse breaking with potholes, so how can I go about it.
Answer: Braking involves all the wheels while a pot hole involves one at a time.  If your phone has a three axis accelerometer, like an iPhone, the acceleration vector will have a different direction profile relative to the phone for each phenomena. You should also expect a shorter time profile, ie higher frequencies for pot holes.
These should be distinguishable in the data you collect, so collect data and answer your own question

Accelerometer to track Potholes


Entropy Change In Rusting of Iron

Question: I presume that the entropy of the system should decrease during rusting of iron because the atmospheric oxygen gets converted to solid state from gaseous state which should result in the decrease of randomness. So am I correct?
Comment: You know very well that you are wrong. Homework: Where do you suspect the entropy goes?
Answer: It is necessary to differentiate between the change in system entropy and surrounding entropy.
Rusting involves a decrease in system entropy, but causes an even greater increase in the entropy of the surroundings. This is because it is an exothermic reaction, releasing heat energy to the surrounding molecules, increasing their disorder.
Overall, when you calculate total entropy (system entropy + surroundings entropy) you find an increase. Hence, the process in spontaneous.

Entropy Change In Rusting of Iron


How to create Destination table name with Getdate in SSIS OLEDB destination

Question: How to create Destination table name with Getdate in ssis destination task.
I have one table like Test..but i want to create new table like Test_09Feb2023 in OLEDB destination.
Comment: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Comment: I want create the new table dynamically with date(tablename_09022023 like.. using source table name in destination server
Answer: For this to work, you need the following pieces.
Variables
You're going to want to use the SSIS Expression language to build out a date string. You would like to see a format of DDMonYYYY. The SSIS expression language has year/month/day functions that return the numeric equivalent. Your task will be to convert the month number to the abbreviation.
Being lazy, I would tell people that the table name will be Test_YYYY-MM-DD because that's what I have right here
(DT_WSTR, 4)YEAR(@[System::StartTime])
+"-"
+ RIGHT("0" + (DT_WSTR, 2)MONTH(@[System::StartTime]),2)
+ "-"
+ RIGHT("0" + (DT_WSTR, 2)DAY(@[System::StartTime]),2)

The important thing to understand is that year/month/day make numbers so you must then cast them as string, thus the (DT_WSTR).
For month or day, those are numbers so it will not return a leading zero. For that, you need to first convert to a string. Then we prepend a zero.
Rather than deal with complex expressions of when to prepend a zero, we always do it and then slice off the last 2 digits. 273/356 times of the year, there's nothing to slice but the logic is easier to maintain if something goes awry.
I would call that Variable DateString of type String.
The next variable I need is the table definition. I click Add, name it TableDefinition of type String. We will again use an Expression.
"CREATE TABLE dbo.[Test_"
+ @[User::DateString]
+ "](test_sk int identity(1,1) NOT NULL,  col1 int, col2 int, col3 nvarchar(128), CONSTRAINT [PK__dbo__Test__"
+ @[User::DateString]
+"] PRIMARY KEY (test_sk));"

Rendered and formatted
CREATE TABLE dbo.[Test_2023-02-09]
(
    test_sk int IDENTITY(1, 1) NOT NULL
,   col1 int
,   col2 int
,   col3 nvarchar(128)
,   CONSTRAINT [PK__dbo__Test__2023-02-09] PRIMARY KEY(test_sk)
);

As I've gone through the whole answer, I realized I could have used and re-used another variable QualifiedTableName  This creates a guaranteed safe name version of our string
"[dbo].[Test_" + @[User::DateString] + "]"

Rendered, that would be
[dbo].[Test_2023-02-09]

Finally, I'd add a third Variable that maybe I only use in development cycles to remove the table. DropTable
"DROP TABLE IF EXISTS dbo.[Test_" + @[User::DateString] + "];"

This uses syntax that is only valid from SQL Server 2017+.
Control flow
Now we can start designing the package itself

The two Execute SQL Tasks will differ only in the value of SourceVariable

Right now, before you do the magic of defining the Data Flow Task, run the SSIS package as-is. This matters because you need to get today's version of the table out there.
Data Flow Magic
Before an SSIS package starts running, it checks that all the resources it needs exist and match the contract it had regarding data types, etc when it was developed. If anything is out of place, the package fails fast so you don't end up with a half-executed pipeline.
We need to tell the Data Flow that it should only perform validation immediately preceding execution. Right click on the Data Flow Task and choose Properties. Change the value of DelayValidation from the default of False to True.
Validation still has to happen, you cannot get an SSIS package to run without eventually performing the check, we have simply delayed it as the preceding task, "SQL Create Table" is responsible for making sure the metadata is valid for the task.
Design the Data Flow task as you need. In the destination, you need to signal that the table name comes from a variable.
If an OLE DB Destination is your sink, then change the "Data access mode"

If your sink is an ADO.Net or ODBC destination, then once they are on the canvas, you will click back to the Control Flow, right click on the Data Flow Task and on the Properties Menu, find Expressions and then the desired Property
ADO.NET Destination demo

ODBC Destination demo

Either way, you'll want to pick our variable @[User::QualifedTableName]  As I think about it, one of these, uses double quotes for a qualified table name. Instead of [dbo].[Table_etc] it might expect "dbo"."Table_etc"
Experiment, play. Those are the basic building blocks you'll need to make this happen specific to your environment and problem domain.
But, I need SSIS to inspect the contents of today's file to determine the table structure
If that is your problem definition, then SSIS is not the tool for you. As noted in the previous section, a data flow must pass validation. I cannot have  a dynamic "shape" to the data. For that, you need a time machine and use DTS or go cloud and use Azure Data Factory.
Comment: I have create the table with date using the variables, but here showing the error  Oledb destination not existing the table.how i fix this issue.
Comment: I'm sorry, I don't understand your comment. Click Edit on your original question and paste in a screenshot of 1) the OLE DB Destination, 2) the Variables used 3) the running package with the errors fully visible
Comment: Also, did you follow the instructions? Specifically `Right now, before you do the magic of defining the Data Flow Task, run the SSIS package as-is. This matters because you need to get today's version of the table out there.`
Comment: Hresult: 0x80004005  Description: "The metadata could not be determined because every code path results in an error; see previous errors for some of these.".
An OLE DB record is available.  Source: "Microsoft SQL Server Native Client 11.0"  Hresult: 0x80004005  Description: "Invalid object name 'dbo.Items_2023-02-09'.".
Error at Data Flow Task [OLE DB Destination [142]]: Failed to open a fastload rowset for "[dbo].[Items_2023-02-09]". Check that the object exists in the database.
Comment: After create table dynamically using variables, but im not forward to select destination  table or  source nd destination mappings is not possible..


What is the origin of vampires completely disintegrating?

Question: There are multiple vampire myths (Buffy and Blade to name two) that include the physical attribute of vampires to completely disintegrate upon final death (no remaining dust, residue or particles, including a lack of clothing).
In what work did this attribute of vampires first appear?
Comment: There was this little known obscure book saying something to the effect of "ashes to ashes, dust to dust"...
Comment: Not an answer but: in *Buffy* specifically, the choice to make vampires dust on death was a practical one: they didn't want to have to explain all the dead vampire corpses lying around Sunnydale. Note that in Whedon's original script (that was butchered into the movie), vampires *didn't* dust upon death, so that was a change he made for the lore of the show.
Comment: seems like this is a "natural" consequence of vampires supernaturally avoiding death and  decay -- when the vampire is destroyed, the body decays suddenly, which in long-buried corpses means becoming a skeleton or even reduced to dust. the burning may represent foreshadowing of eternal punishment in Hell.
Answer: The answer is in Bram Stoker's Dracula. Dracula turns to ashes in the end of the book. A lot of vampire rules come from Stoker. He did extensive research into the vampire myths and legends before writing his book.

But, on the instant, came the sweep and flash of Jonathan's great
  knife.  I shrieked as I saw it shear through the throat.  Whilst at
  the same moment Mr. Morris's bowie knife plunged into the heart.
It was like a miracle, but before our very eyes, and almost in the
  drawing of a breath, the whole body crumbled into dust and passed from
  our sight.
Comment: A lot of vampire stories were word of mouth stories at the time. They weren't written down. So the research Stoker did was in some cases the first documented example. Especially in the case of a vampire turning to ashes or dust when killed. I'll see if I can find any stories written before his.
Comment: You say that Bram Stoker researched various myths, legends and lore to write his book. Does that mean that BS' Dracula the first occurrence of this particular bit of lore, or did it appear even earlier in something he read (or was told)?


How can I give the object the shape I want?

Question: I want to fit the net behind the goal but I don't know how to shape the net object? Example in the picture below (how can I give the shape I want to the black drawn place?);
Comment: Usually this would be done in a 3D Content Creation tool, like 3DS Max, Maya, Blender, etc. and then the resulting model file is imported into a game engine like Unity. Game engines typically don't have such rich 3D mesh editing tools built in, though you can find assets or write custom scripts to achieve some of those behaviours.
Comment: I will take into account what you have said. Thank you.
Answer: I guess that's Unity.
Unity has a tool called ProBuilder that allows you to do basic modeling things. This tool has an option called Probuilderize, which basically converts meshes into editable models through ProBuilder. I've never used this option so I can't talk much about it, however I know it's there and I know it does that.
This is a simple task, so I don't see why not take advantage of what you have on hand. However, I would recommend solving these types of problems in dedicated software, such as Blender. But that is already out of the scope of this answer.
Comment: Thank you, I will try as soon as possible.


Need help retrieving bitcoin from an old phone

Question: I recently replaced my iPhone.  A friend tried to erase the old phone to use it for himself but was told it still had a live bank account - I assume my bitcoin account.  How can I retrieve my account and bitcoin?  I still have my old phone but not much else - anything I can do?
Comment: An app on your old iPhone could not have prevented it from being erased. It was likely your Apple ID account on the old iPhone that prevented it from being restored. Regarding the bitcoin, which wallet were you using on the old phone? Luckily you still have it, because then you can do a backup and use that backup on the new phone. Also, depending on the wallet, there might be a seed phrase (usually 12 or 24 simple words) that if you possess you will be able to resurrect the wallet on the new phone.
Answer: A bitcoin app would not stop your phone from being formatted, so it must be something else.
That said, if you still have bitcoin in a wallet on your old phone, then you should definitely transfer it to a wallet on your new phone (or maybe just recreate your old wallet on your new phone). Just open up the bitcoin wallet, and transfer the funds / export the seed phrase.
Retrieving the bitcoin won't fix your problem of being unable to format the phone, but you should definitely not sell your phone until you have done so!