Sum of inverse of Fibonacci numbers

Question: If $F(n)$ is the nth Fibonacci number, How can I prove that:
$$\sum_{i=1}^{\infty} \frac{1}{F(i)}\approx 3.36\, .$$
Comment: If all you need to prove is an approximation, you can simply compute the sum of the first $N$ reciprocals. For $N=14$ you get $3.36$.
Comment: This may be interesting: http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html
Comment: @EricS. Unless you estimate the remainder, this proves nothing.
Comment: See https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant
Comment: Related: [What is the sum of Fibonacci reciprocals?](http://math.stackexchange.com/questions/732155/what-is-the-sum-of-fibonacci-reciprocals)
Comment: Your question is a bit unclear, but if you really want to show that this is a good approximation, you should probably add ([tag:approximation]) tag.
Comment: @Jean-ClaudeArbaut. Hmm, right you are. My bad.
Answer: Since $F(n) \approx \frac{\phi^n}{\sqrt 5}$ for large enough $n$, you may use that as an approximation, which give you a geometric series. For instance, if we use the approximation from the fifth term on, we get
$$
\frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \sum_{n = 4}^\infty \frac{\sqrt 5}{\phi^n} \approx 3.3612
$$
To get a feel for how accurate this is, $F(5) = 5$, while $\frac{\phi^5}{\sqrt 5} \approx 4.96$, which is a relative error of less than $1\%$. The relative error gets smaller and smaller as $n$ grows, by a factor of about $3$, and it is alternating (every other term is too large, every other term is too small) which also reduces the error.
Since every term in the sum is less than $1\%$ away from the corresponding term in the original sequence, the true answer is within $1\%$ of the sum we have, which is $\sum_{n = 4}^\infty \frac{\sqrt 5}{\phi^n} \approx 0.53$. Thus the error of this approximation is at most about $0.005$. One more exact term, and the absolute error will be at most $0.001$.
Comment: @Amin23 What do you mean "proofing"? Do you mean calculating $\sum_{n = 5}^\infty \frac{\sqrt 5}{\phi^n}$? Then the answer is "yes", and the exact value is $\frac{\sqrt 5}{\phi^4(\phi - 1)}$. For more details, you can google "sum of convergent geometric series", it will tell you much more than I can tell you in a comment.
Comment: @Amin23 So what you really want is not to prove that $\sum_{i = 1}^\infty 1/F(i) \approx 3.36$, but you rather want to _calculate_ the sum $\sum_{i = 1}^\infty 1/F(i)$? Then I have no idea, and that page is probably your best bet.
Comment: Thanks for useful comment. Is there a simple analytical method for proofing this summation?
Comment: @Jean-ClaudeArbaut The error is less that $1\%$, as I said in my answer (because every term is less than $1\%$ away from the corresponding term in the true sequence), so the absolute error is at most $0.005$. The error of the next term is about $0.3\%$, so if you want better accuracy you can just add more terms.
Comment: @Arthur I find it it's  analytical method in this site   mathworld.wolfram.com/ReciprocalFibonacciConstant.html
Comment: @Amin23 If you understand everything on that page (the $q$-polygamma function, the Lambert series and the Jacobi elliptic function), then go ahead. I don't, so I will stick to this.
Comment: @Arthur Just because of these functions that you mentioned I ask you a simple analytical method, But you misunderstanding my meaning.
Comment: @Arthur Ok,Thanks.
Answer: $$
\begin{align}
F_n
&=\frac{\phi^n-(-1/\phi)^n}{\sqrt5}\\
&=\frac{\phi^n}{\sqrt5}\left(1-\left(-\frac1{\phi^2}\right)^n\right)
\end{align}
$$
Therefore,
$$
\begin{align}
\sum_{n=1}^\infty\frac1{F_n}
&=\sum_{n=1}^\infty\frac{\sqrt5}{\phi^n}\left(1+\left(-\frac1{\phi^2}\right)^n+\left(-\frac1{\phi^2}\right)^{2n}+\left(-\frac1{\phi^2}\right)^{3n}+\cdots\right)\\
&=\sqrt5\left(\frac1{\phi-1}-\frac1{\phi^3+1}+\frac1{\phi^5-1}-\cdots\right)\\
&=\sqrt5\sum_{k=0}^\infty\frac{(-1)^k}{\phi^{2k+1}-(-1)^k}
\end{align}
$$
Since $\frac{\sqrt5}{\phi^{19}+1}=0.0002392$,
$$
\sqrt5\sum_{k=0}^8\frac{(-1)^k}{\phi^{2k+1}-(-1)^k}=3.3600587
$$
is less than $0.0002392$ too high.
Comment: @Amin23 : Yes.  That bound is zero, uniformly in $n$.  https://en.wikipedia.org/wiki/Golden_ratio#Relationship_to_Fibonacci_sequence
Comment: As $n$ gets larger, that error tends to $\frac1{\phi\sqrt5}$, it does not vanish.
Comment: It is good approach for approximation of exact value of  this summation. Thanks
Comment: +1 This is one of the good ways how to estimate the error.
Comment: Is there a bound or function based on parameter $n$ for this approximation $\mid F(n)-\frac{\phi^n-{(-1/\phi)}}{\sqrt{5}} \mid$

Sum of inverse of Fibonacci numbers


Travel to Japan in December 2020

Question: Before COVID-19 occurred, I was planning on going to Japan in December 2020. However, with things escalating, I decided to not give in and book a flight when I almost did. I know Japan has closed their border from the United States. I was looking at flights in December and the prices are really enticing. I wanted to know if I can book a flight for December 2020. It is kind of risky given the idea of when Japan will open their border from the United States. I wanted to know if there is a way for me to get a potential full refund or about 75% back if I find out at the end of November 2020 that Japan is not opening their borders? The flight is Air Canada.
Comment: I think you could be able to buy travel insurance with a Cancel for Any Reason clause.
Comment: Can you suggest such a policy that doesn't explicitly exclude issues related to the pandemic? I can't see any insurance company taking that on!
Comment: FWIW, it's extremely unlikely that you will be able to enter Japan as a tourist in December.  They've yet to open any "green lanes" for business travellers, and until very recently even permanent residents were not allowed to return.
Answer: We can't say when Japan will open their borders. If the flight you book departs but you can't fly because you don't qualify to enter Japan then the airline policy is clear: it's your problem and no refund will be offered.
You can, of course, book a fully refundable ticket but not, I suspect, at the price that's tempting you right now.
Comment: Some airlines are offering more refundability/changeability than usual on their inexpensive tickets, so it is worth reading the website of the airline you're considering flying to see if you can, in fact, get a ticket which is sufficiently refundable for a tempting price. (Of course, if the airline goes out of business, all bets are off.)
Answer: A lot of airlines have made provisions for their customers in case the Covid-19 restrictions of various countries impede with their travel plans. A quick search on AirCanada's website shows that they too have implemented Covid-19 goodwill policies – likely because they prefer a booking in hand than not. The exact text reads:

We have implemented highly flexible and expanded booking options. You can make a one-time change without a fee for all new or existing bookings made through September 30, 2020 for original travel between March 1, 2020 and September 30, 2021. If you booked directly with Air Canada and you need to cancel for any reason, you can convert your ticket to an Air Canada Travel Voucher that has no expiry date or to Aeroplan Miles with an additional 65% bonus miles.

Therefore, as it currently stands you can book your flight in December as planned (until 30th September) and if it becomes obvious that Japan is not sufficiently opening its borders you can choose whether to change the dates without a fee or get a travel voucher refund.
Do note that Air Canada is likely under no legal obligation to provide this service except as stated in the terms and conditions of your actual booking (the contract). Most importantly, there may be clauses asking you to pay a price difference between your original flight and the one you change to; and Air Canada will most likely refuse to give you a cash refund rather than a voucher in the event that you cancel (except if the ticket itself includes cancellation – but then it is unlikely to be cheap).
I am sure that the agents at Air Canada's hotline or other members of their customer service team will be happy to answer any additional questions about this policy that you may have – specifically concerning the caveats I mentioned above.

Travel to Japan in December 2020


how to create .doc file using java?

Question: Can anyone tell me how to create doc files using java?
I know, there's a POI library, but it seems like it can save only simple documents. You can read anything you wish, but you can't save it all back again. Or may be i missed something? How can i save whole document with pictures, tables and styles?
Comment: Duplicate: http://stackoverflow.com/questions/203174/whats-a-good-java-api-for-creating-word-documents
Answer: Docmosis lets you do heaps of styling easily via the template.  Uses OpenOffice and Java to do the job.  Its free and free to distribute.
Answer: It ain't cheap, but you could try Aspose.Words. It'll do everything you state above and more.
Answer: JODConverter will allow you to convert lots of different document formats
Answer: OpenOffice.org's Universal Network Objects (UNO),  allow you to generate .doc,.PDF,  as well as OpenOffice documents. It supports several programming languages like: Java, C++, Visual Basic, etcc..
Some good things is that: its free, open source and plataform-independent
You can build documents, spreadsheets, presentations, etc. Start from scratch or using a template and fill the gaps..
In order to use it you will need to include some libraries that comes with the OpenOffice suite.
Useful links:

Open Office home
Open Office UNO Developer's Guide

how to create .doc file using java?


How to install virtualbox on Ubuntu 20.04?

Question: So I have been trying to install virtualbox on my live usb with persistent storage. I have about 100GB so I know that is not the issue, I have followed several youtube video through the process of installing virtualbox, but no matter how hard I try I cannot get virtualbox to ever install. I download the virtualbox software from the website, then I open terminal in downloads, run sudo dpkg -i install <name of file> and then run sudo apt-get -f install but instead of installing dependencies, it just forces me to uninstall the virtualbox installation. The files doesn't disappear from my downloads, but no matter what it will never download the dependencies.
Comment: It's in the [repos](https://packages.ubuntu.com/search?keywords=virtualbox). You should be able to install it with `sudo apt install virtualbox`.
Comment: Does this answer your question? [How do I install a .deb file via the command line?](https://askubuntu.com/questions/40779/how-do-i-install-a-deb-file-via-the-command-line)
Comment: It's possible the virtualbox package you downloaded was for a different OS or release, and thus the only fix is it's removal due to dependency clash.  We can't help with that without having access to the actual message lines on your install commands (initial install and/or --fix-broken).  Why not use the Ubuntu repo. version as KGIII said!
Comment: Does this answer your question? [How to install VirtualBox from command line?](https://askubuntu.com/questions/367248/how-to-install-virtualbox-from-command-line)
Comment: I did try to do 'sudo apt install virtualbox' from outside of the Downloads directory.

Here's what it gives me:

```Package virtualbox is a virtual package provided by:```

```virtualbox-6.1 6.1.16-140961~Ubuntu~eoan```

```virtualbox-6.0 6.0.24-139119~Ubuntu~eoan```

```You should explicitly select one to install.```

```E: Package 'virtualbox' has no installation candidate```
Then I ran this:

```sudo apt install virtualbox-6.1 6.1.16-140961~Ubuntu~eoan```
result:

```E: Unable to locate package 6.1.16-140961~Ubuntu~eoan```
Comment: Just do sudo apt install virtualbox-6.1, leave off the rest
Comment: I have had best luck installing VBox from the Ubuntu Software icon.
Answer: Seems like you are describing the correct, expected behavior.
dpkg does not handle dependencies or repositories. It only installs or uninstalls the package that it's told to (that's an oversimplification).
Instead, most folks use apt when possible. Apt does handle repositories and dependencies.

You can use apt to install downloaded packages: sudo apt install /path/to/file_name.deb
Generally, it's easier to use the Ubuntu repositories than to download software yourself: sudo apt install virtualbox

Opinion: I've used Virtualbox every workday for years. Most users do not need to expend the extra effort to download Vbox from upstream and puzzle out the dependencies. The Ubuntu package works quite well, is already meshed to the correct dependencies, and is fully-featured.
Comment: A stock Ubuntu 20.04.x system does not look for packages labelled "eoan". It's that simple. We don't know what changes you made to convince your system to look for a wrong-version non-Ubuntu package (nothing resembling that package version string is on http://packages.ubuntu.com). Whatever you did, undo it. If you don't know how to clean up previous failed attempts, then re-make your USB to wipe those changes.
Comment: Alright so I tried ```sudo apt install virtualbox``` Here's what it gives me:

"Package virtualbox is a virtual package provided by:

virtualbox-6.1 6.1.16-140961~Ubuntu~eoan

virtualbox-6.0 6.0.24-139119~Ubuntu~eoan

You should explicitly select one to install.

E: Package 'virtualbox' has no installation candidate"

Then I ran this: ```sudo apt install virtualbox-6.1 6.1.16-140961~Ubuntu~eoan```

result: "E: Unable to locate package 6.1.16-140961~Ubuntu~eoan "
Comment: Looks like you have a 19.10 (Eoan) system instead of a 20.04 system., 19.10 is  not supported anymore, and we don't answer questions about it anymore. Consider migrating to a supported release of Ubuntu. If you are indeed running 20.04, then you have unwisely added 19.10 sources and/or packages. Remove them.
Comment: I just burned this Live USB earlier today, so I am certain that I have not made any any changes to the sources of packages. I was also almost certain that it was 20.04, I used the latest stable release from the website. How can I determine if my sources are messed up or if I have the wrong version?     EDIT: My Settings -> About says: Ubuntu 20.04.1 LTS


Does the SHA-256 sum of a 256-bit key help you crack it?

Question: If I have the SHA-256 sum of a 256-bit AES key, can I determine the key any faster than if I merely had a ciphertext and the corresponding plaintext?
Answer: The practical answer is "not really". With that said, here's a more nuanced answer:
If you only have a single AES-256 plaintext/ciphertext pair (one block, that is), it's not actually possible to bruteforce the key. AES has a block size of 128 bits, so the probability of any given key having that plaintext/ciphertext pair should be $2^{-128}$. That number is quite small, except for the fact that there are $2^{256}$ possible AES-256 keys, so there should be about $2^{128}$ keys with this plaintext/ciphertext pair. In other words, even a bruteforce search would still leave us with about $2^{128}$ possible keys. With access to a SHA-256 hash of the key, we can instead search for a preimage of the hash, which should leave at most a few possible keys (those keys can then be checked against the AES plaintext/ciphertext pair).
So, in theory, the hash should help an attacker in the above situation. Of course, in practice, it's impossible to search a key space of $2^{128}$, let alone $2^{256}$, so the above has no practical implications.
A more interesting question is whether an attacker who knows both $\text{SHA-256}(k)$ and $(m,E_\text{AES-256}(k, m))$ can combine that knowledge in order to use some more efficient method.
Here's an example of how this kind of problem could happen. Suppose we have two secure cryptographic hash functions $f$ and $g$, each operating on 256-bit inputs and producing a 256-bit output. Let $f(m) = \text{SHA-256}(m)$ and $g(m) = f(m) \oplus m$ (where $\oplus$ is XOR). I haven't checked this example carefully, but $g$ should still be preimage resistant. If an attacker is given either $f(k)$ or $g(k)$, then they should be unable to determine $k$ without a brute force search. However, if the attacker is given both $f(k)$ and $g(k)$, then they can compute $f(k) \oplus g(k) = f(k) \oplus f(k) \oplus k = k$ to recover the original input.
As far as we know, there aren't any interactions like this between SHA-256 and AES-256. This kind of situation definitely deserves caution, since as far as I know, the algorithms were not explicitly designed to avoid interaction between the two. That said, there does seem to be a fair amount of confidence that there will not be any surprising relationships discovered between AES-256 and SHA-256.

In most cases, it's better to derive separate keys for separate uses. Instead of using $k$ directly, use $k$ to compute two subkeys $k_1$ (for AES-256) and $k_2$ (for SHA-256). There are a variety of ways to do this. My favourite is to use HKDF (that RFC is surprisingly readable), which might look like this:
$k_1 = \text{HKDF-Expand}(k, \text{0x01}, 32) \\
k_2 = \text{HKDF-Expand}(k, \text{0x02}, 32)$
Comment: Excellent answer!  For context, I've written a system for sharing bits of sensitive data, where each piece of data is encrypted with 256-bit AES and then stored, with a randomly-generated identifier pointing to it.  Thus, to share the secret, you must send the identifier and the key to the recipient.  Somebody suggested I use the SHA-256 sum of the key as the identifier instead, reducing the amount of data that needs to be pasted to/from a chat window.  I was concerned that there could be an interaction between the two algorithms like you described.
Comment: @waucka If your still worried about too much info leaking, you can use just the first 4 or so bytes of the hash as your identifier, which will increase the number of possible keys matching by quite a large amount.
Comment: That is an excellent idea.  Since the data is deleted from the server after 24 hours (and the server won't get tremendously heavy use), I really don't need the full 256 bit space for uniqueness.
Answer: yes, in the same way the hash of a password does help you recovering the password.
If you have the hash you can test against it. This doesn't mean that the recovery is doable in a reasonable amount of time
Comment: The hash of a password (in the sense hash has in the present question) often allows to recover the password! There are rainbow tables for that.


Effect of base doping on early voltage,base transport factor and emitter injection efficiency?

Question:

According to me as base doping increases recombination will be less at base region so base transport factor and emitter injection efficiency will increase and as effective base width will decrease due to less doping at the base region so early voltage will decrease.
am I thinking right? No options is matching.
Comment: Why does MORE base doping reduce the base recombination?
Answer: First of all you have a typing error in your analysis,
"According to me as base doping increases ..."
Here, it should be decreases and rest are fine;
And yes your conclusion is right ,
i.e,
$$ \text{as } \quad  N_B \downarrow \quad \to \quad \gamma \text{ (Emitter Injection Efficiency) } \uparrow  \quad, \quad \beta^{*} \,\, \uparrow \,\,\& \,\,|V_A| \,\,\downarrow $$
Answer: Yes, lower base doping:
Increase emitter injection efficiency (ratio of emitter:base doping)
Increase base transport factor as lower doping means less recombination for minority carriers injected from the emitter
Increase early voltage - because the base-collector junction depletion region will deplete more into the collector than the base- therefor the base edge of the depletion region will not reduce the overall base width as much.


Can I deploy Juju on Eucalyptus?

Question: Please let me know if I can deploy juju on Eucalyptus, if yes please provide me the configuration details which i need to edit on environments.yaml file.
Answer: As long as eucalyptus provides ec2 API, you can use juju to deploy your workload on it.
And config is similar to ec2's. Modify endpoint accordingly. Note that we do not routinely test on Eucalyptus, but if you want to provide a working configuration snippet you can send it to the Juju mailing list for consideration of including it in the documentation.
Comment: Thank You very much, I will try to deploy and let you know the status
Comment: How did it go? If you have a Eucalyptus cloud handy we'd be interested in setting up an automated test so we know Juju is working with Eucalyptus clouds, and fix it if that inadvertently changes.
Comment: I have tried by editing environments.yaml file with eucalyptus details. but i cant run bootstrap command as i am getting the below error error: cannot parse "/root/.juju/environments.yaml": YAML error: line 30: did not find expected key. Please help me
Comment: @Mark Shuttleworth - I have a Eucalyptus Cloud handy

Can I deploy Juju on Eucalyptus?


How do I share apps with my kids in Android 10?

Question: I upgraded the family tablet from a HUDL2 to a Teclast P20HD. In the HUDL (Android 4.4.2) when I bought/installed new apps I was given the option to share them with other (restricted) users on the tablet. When I clicked on the user in the user settings, I got a list of installed apps that could be enabled/disabled for each user (my kids). This system was simple and effective.
With the Teclast (Android 10) I only have the option to add and delete users. No app control whatsoever.
I setup a Google account for my youngest (5yo) which I can manage through family Link (as I do with my older child's devices) but then there is no way to share apps from my own Google account to his (the family library feature is only for teenagers and won't cover the apps I've already purchased anyway). I have purchased dozens of kids apps over the last couple of years and I obviously want these available in my son's account.
I tried sharing apps with the Family Link feature inside the play store app on my account, but of the 100+ apps I've purchased, only 19 got shared.
I tried adding a second Google account from inside my son's tablet user space (my own Google account, so I could get the apps again on the play store with my credentials), but it won't let me; expecting all further accounts inside that user to be my kid's school accounts and such.
So I'm stuck. Is it possible to get the kids apps I have purchased into my son's user space on this tablet?
If not, is it possible with some other tablet (I could send this one back and get something different) or is the KitKat way of things now deprecated and I'm stuck with having to re-purchase hundreds of pounds of kids apps?
Comment: The app sharing system of your old tablet was something proprietary added by the manufacturer of the tablet. Your new tablet doesn't have such a modification, the multiple-user system is just plain Android.
Answer: Robert is correct. Damn. I tried multi-user on different device and it's just as plain.
For anyone coming here from Google, I have further information and a couple of solutions..
Firstly, you can now setup child accounts at google for under 13s and, once added to the family group, they can legitimately access apps from your main account, so long as you setup Family Library and manually enable it for each app on the app's play store page (as mentioned, only 19 apps came over automatically when I first enabled this - I mistakenly believed this was the sum total of eligible apps). This is tedious but works but once enabled, future purchases can be added automatically (in your Family Library preferences at the Play Store).
This also works for apps that were "purchased" for free, during sales in the last five years.
Note: At the time of writing, setup of Google Family Library still states that users must be over 13 to receive apps, but this is evidently no longer true.
Before realising all this, I deleted my son's Google account from the new Android user, then used my own Google account to quickly install all the apps (from my Play Store apps list). Once done, replaced my account with my son's and we're in business.
Except with a different tablet. This rubbish-with-no-sensors is going back. If you came here considering buying a Teclast P20HD, I recommend you reconsider.


Why isn't there a female Shankaracharya in Hinduism?

Question: Why are not women made pontiffs in Hinduism or in any other religion for that matter?
Comment: Not sure about other cultures, but in Hindu religion, nothing is stopping them, If they show the required characteristics. There had been some female saints though like "Mata Anasuya", https://en.wikipedia.org/wiki/Anasuya
Comment: Also, see [Why are women not allowed to chant the Gayatri mantra?](https://hinduism.stackexchange.com/questions/9200/why-are-women-not-allowed-to-chant-the-gayatri-mantra)
Comment: Sharada Devi https://en.m.wikipedia.org/wiki/Sarada_Devi
Comment: can you specify what you want to know exactly.."pontiffs" doesn't seem to be right term in hinduism..did u mean female saints/scholars..if so then, I think there are many examples like Maitreyi, Gargi apart from what pointed out in above comments..
Comment: Adi Shankaracharya mentions in Mathamnay (constitution for 4 Shankara mutts) that only Samnyasis can ascend the position of Shankaracharya. Women cannot enter the 4th order of Samnyas ( as per all Orthodox Hindu Sampraday) Hence they can't become Shankaracharya.
Answer: Three letters written by Sri Vaishnava acharyas opposing ISKCON's decision to appoint female diksha gurus illustrate why Hindu women, in general, cannot become chief priests, acharyas or shankaracharyas. The central issue revolves around whether there is scriptural sanction or established tradition that allows women to initiate others.
Muralidhara Bhattar, Chief Archaka of Sri Ranganatha Swami temple, in his letter, writes:

After reviewing these statements, my conclusion is that the introduction of the female Diksa Guru system in ISKCON will set a bad example to the society of followers of Sanatana Dharma, and it will certainly have wider ramifications for other sampradayas.
Protecting our own sampradayam may include, and is not limited to, withdrawal of the 1970 letter issued by my late father, Sriman Rangaraja Bhattar, on behalf of the Sri Vaishnava sampradayam, certifying that ISKCON is a genuine Vaishnava sampradayam.
As a friend and well-wisher of ISKCON I urge the GBC body to reject this proposal once and for all. Introducing such ashastriya practices will bring about only yoga nashta to the institution.

In another letter to ISKCON, Professor M. A. Lakshmithathachar, the presiding acharya of the Anantarya Peetham, Melkote, writes:

I am deeply concerned with the turmoil that is going on with regards to the mantropadesa and initiation by women preceptors in ISKCON. In a growing organization like ISKCON, confusions, turmoil, statements for and against are bound to prop up. Senior persons at the helm of affairs should go through our ancient texts and try to find out solutions for such problems.
...
Presently, in the case of permitting women acharyanis, the same procedure has to be adopted. Whatever may be the statements of the shastras, either permitting or prohibiting the acharyanis to do mantropadesa or not, the decision should solely depend upon shistacara. In the history of ISKCON if there is shistacara of mantropadesa by women then acharyanis are free to do mantropadesa. There could be some exceptional cases which might have happened due to the demands of time (āpatkālenāstimaryāda). Exceptions should not become the general rule. Since there is no crisis as to introduce acharyanis to initiate, this should not be considered a serious issue at present.
If ISKCON, an internationally famous organization, goes against the shistacara and be successful, probably the other schools of Vedanta like Ramanuja, Vallabha, Nimbarka, etc., may follow suit, which results in the practice of violating the shishtacara and is not desirable. I wish ISKCON, an internationally reputed organization, will not establish a new shistacara violating all the shistacaras practiced even today in all the schools of Vedanta.

You can read the third letter here.
Interestingly, ISKCON appears to have ignored all these letters and went ahead with their initial decision citing its founder Prabhupāda's commentary on this topic:

Srila Prabhupada severely criticized and warned against bodily and mundane considerations in regard to the role and eligibility of diksa-guru — whether in the form of casteism, selfish motives for profit, adoration or distinction, or as conformity to social, cultural or modern trends and artificial "equality":

...Generally a spiritual master who constantly instructs a disciple in spiritual science becomes his initiating spiritual master later on. (...) Sri Jiva Gosvami advises that one not accept a spiritual master in terms of hereditary or customary social and ecclesiastical conventions. One should simply try to find a genuinely qualified spiritual master for actual advancement in spiritual understanding. (Caitanya-caritamrta, Adi-lila 1.35 purport)


smooth irreducible variety with finite (non-zero) Picard group?

Question: Does there exist a smooth irreducible variety $X/\mathbb{C}$ such that $\mathrm{Pic}(X)$ is finite and non-zero?
Answer: Yes: take $\mathbb{P}^2_\mathbb{C}-V(f)$, with $f$ degree $d$. Then, this is smooth and irreducible and has Picard group $\mathbb{Z}/d\mathbb{Z}$.
This is not true if you require $X$ to be projective, then $\mathrm{Pic}(X)$ contains $\mathbb{Z}$ if it's non-zero.
EDIT(Explaining what happens if $X$ is projective): Suppose that $X$ is not affine, and so not $\text{Spec}(\mathbb{C})$ (I'm assuming $X$ is integral, just for convenience). Then, there is a very ample line bundle $\mathcal{O}(1)\in\mathrm{Pic}(X)$. But, any power of $\mathcal{O}(1)$ is still very ample, and since $X$ is not $\text{Spec}(\mathbb{C})$, $\mathcal{O}_X$ can't be very ample. So, $\mathcal{O}(1)$ can't be torsion. So, we see that $\mathbb{Z}\subseteq\mathrm{Pic}(X)$.
In fact, you can say more. I'm going to restrict to the case of $X$ a (smooth projective integral) curve, purely for convenience again. Then, we know that $\text{Pic}^0(X)$ are the $\mathbb{C}$-points of an abelian variety (the Jacobian) of dimension $g=\text{genus}(X)$. So, if $X\ne\mathbb{P}^1$, then $\mathrm{Pic}(X)$ is uncountable.
Comment: thanks! can you say why $\mathrm{Pic}(X)\neq 0$ needs to contain $\mathbb{Z}$ in the projective case? if the answer is too long, I can post as a separate question.
Comment: @adrido I've edited my response.
Comment: Excellent answer, dear Alex. Maybe for the sake of clarity you could make explicit that your Edit addresses the projective case.
Comment: @GeorgesElencwajg Thanks :) I've edited for clarity.
Comment: Dear Alex, thanks for the clear explanation.
Comment: what about connected smooth proper non-projective $\mathbb{C}$-schemes?


How much do graduate schools in Mathematics care about the Statement of Purpose?

Question: Can a 'bad' SOP keep you out of a program? Can a 'good' SOP help you? Can the SOP only hurt you?
Comment: Too variable for a general answer. Every university is different. Every reader is different. I advise you to write a good one, in any case.
Comment: Related:https://academia.stackexchange.com/q/81486/101
Comment: See https://academia.stackexchange.com/questions/38237/how-does-the-admissions-process-work-for-ph-d-programs-in-the-us-particularly, the section on "statement of purpose".
Answer: [I have been on the faculty committee for Mathematics Graduate Admissions at the University of Georgia for seven admissions cycles, three times as the head of the committee.]
I view the statement of purpose as one of the least important parts of the application.  It is certainly a part that has very little correlation with acceptance.  In other words, if we look only at, say, the GRE scores or GPA of an application and see that they are extremely high (respectively, extremely low), we can significantly raise (respectively, significantly lower) the chance that that application will be accepted compared to the mean acceptance rate, but a really excellent or really awful personal statement does not change the probability appreciably unless further conditions are met.
Why is this?  The main reason is that, for a PhD student of (most likely theoretical, in our department) mathematics, good or bad statements of purpose correlate too weakly with the qualities for which we are trying to select.  Good writing skills are important, but because a "statement of purpose" is a genre with which most applicants are completely unfamiliar, how they write in this genre doesn't tell us that much about their general writing skills.  (Imagine that we asked them to contribute a haiku or sonnet instead.  That sounds ridiculous...but actually does give some idea of the situation.)
The majority of statements of purpose we get mostly try to convey the student's ardor for mathematics, beginning with an anecdote about how it was kindled at an early age.  They go on to say something about what kind of mathematics they want to study...and in most cases they will show that they don't really know.  As far as our reactions to this: (1) well, we want students who love mathematics, but that is shown better through anything other than how eloquently they write an essay about this love and (2) it is actually okay that most students of (again, mostly theoretical) mathematics have quite vague ideas about what they want to study.  My thesis for instance was on abelian surfaces with quaternionic multiplication.  This is a topic in arithmetic geometry, which is a subfield of number theory.  I believe that in my statement of purpose I said I wanted to do number theory.  Good guess!
I will come back later and say more about (1) Why most departments require a statement of purpose anyway and (2) What are some situations in which a statement of purpose becomes more important.  I wanted to leave an answer before the question got closed.  (It is a good question.)
Added: As Mark Meckes points out in the comments, I have already answered a similar question on this site.
Comment: Interesting.  The statement of purpose is one of the **most** important components of graduate application in computer science, including in subfields of theoretical computer science that are indistinguishable from, if not actually, mathematics.  We don't look for evidence of _ardor_; if they didn't like the field, they wouldn't apply!  We look for evidence of intellectual maturity, independence, creativity, and _technical_ writing ability.
Comment: In addition to being the least important criterion, could you comment on whether its effect is asymmetric? I've heard that it is essentially impossible to get a big boost from a great SOP but it's certainly possible to have a bad SOP sink an entire application. The OP asked specially about this asymmetry.
Comment: I will write more later...but yes, I think it is easier to say "exactly the wrong thing" than "exactly the right thing" in a SOP (though we are happy to ignore a lot of things in SOPs that we are not that thrilled about, as I alluded to above).
Answer: All of the above. Of course the Statement of Purpose is important, or the university would not ask you to write one. It costs time for an evaluation committee to read through every application, and if there were a better way to judge the quality of a candidate, we would use it.
So yes, a bad SOP can totally sink an application. A good SOP can definitely help; it can also explain other issues that a reader might find in your application, such as poor grades or gap years.


Drupal: Custom "Action" Button for nodes On a View

Question: I have a view listing different "task" nodes called "available tasks"
I wish to have a button for every listing that will enable the user to assign the "task" to themselves.
My "task" node has a user reference field called "assigned to".
So all i need is a button next to every "task" node in my "available tasks" view clicking on which will set the user reference field "assigned to" of the "task" node to the current user.
If any one can point me in the right direction; it will be greatly appreciated.
Thanks
Answer: I would recommend the Views Bulk Operations module. With that you can add a 'Display' to your view, then set its 'Style' under 'Basic Settings' to 'Bulk Operations'. Assuming you already have a 'Page' display on your view which lists out your content, then you can attach the new VBO display to your page.
You will need to configure your VBO display to allow 'Modify node fields', then choose your user reference field from your content type as the field to modify.
When that is done, save your view, then navigate to your page. You should then see a checkbox next to each item in your view, and a button that will allow you to bulk modify the user reference field on whichever nodes are selected.


Do questions about vehicles security belong here?

Question: As in questions about cars doors, locks and keys.
Answer: In short yes.

Questions related to the maintenance and repair of motor vehicles and
  their accessories, internal combustion engines, even in applications
  other than automotive so long as the question is related to engine or
  engine accessories.

See the help center for more information.
Answer: Yes they do but....
I'm reluctant to get too detailed.
I feel like we get trolled by theives from time to time trying to find ways to defeat security systems.
I'm reluctant to give full details and typically answer with, get your ID, your title and VIN...goto the dealer and have a new key made...or whattever.
I never feel 100% about those questions and they make me nervous.  I don't want to contribute to the knowledge base of someone that may be trying to do harm so most of the time I avoid a direct answer.
Comment: Agreed. Some Fora specifically ban security questions for exactly that reason. I'd say that questions about maintaining the security side are allowed, but not those regarding circumventing it
Comment: The Ford Ranger question yesterday was a good example... I went for 'take it to the dealer'
Comment: Modern automobile security system's aren't like early 80's to mid 90's security systems and require encoded messages between the antitheft module and the PCM. It would probably require more effort and time on the part of the thief to circumvent this. If you look at modern aftermarket remote starts they require a transponder (on keyless start) be left in the vehicle to even start the engine.


Why is the dot product of perpendicular vectors zero?

Question: I've read that taking a dot product is just projecting one vector on the other, so a perpendicular vector will have no components in the other vectors direction. But shouldn't this leave the length unchanged so it has its original magnitude like multiplying it by 1?
Comment: "taking a dot product is just projecting one vector on the other" What?
Comment: When does the cosine rule reduce to the Pythagorean theorem?
Answer: $$u.v=|u||v|\cos  \theta $$
If $ \theta =\pi/2$ we have $\cos \theta =0$
Thus perpendicular vectors have zero dot product.
Answer: To clarify, the projection of $\vec u$ on $\vec v$ is the vector
$$\left(\frac{\vec u\cdot \vec v}{\|\vec v\|^2}\right)\vec v = \left(\frac{\vec u\cdot\vec v}{\|\vec v\|}\right)\frac{\vec v}{\|\vec v\|}.$$
The dot product is a scalar quantity. But the length of the projection is always strictly less than the original length unless $\vec u$ is a scalar multiple of $\vec v$.
Answer: The dot product is that way by definition, this particular definition gives the expected Euclidean Norm. A consistent dot product can be and is defined differently, for example in physics & differential geometry the metric tensor is solved for and ascribes a different inner product at every space-time coordinate, which is the means for modeling curved spaces.
Answer: Visually, nonzero $u, v \in \mathbb{R}^n$ are perpendicular iff the smaller (in $[0, \pi]$) angle between them is $\pi/2$.
With $u \cdot v$ defined as $\sum_{i = 1}^n u_i v_i$, the the law of cosines implies that
$$u \cdot v= \| u \| \|v\| \cos \theta $$
If $ \theta =\pi/2$, then $\cos \theta = 0$, whence $u \cdot v = 0$; perpendicular vectors have zero dot product.
Conversely, if $u \cdot v = 0$, then multiplying by $\frac 1 {\| u \| \| v \|}$ gives $\cos \theta = 0$, so the angle between $u$ and $v$ must be $\theta = \arccos 0 = \pi/2$.
In conclusion, two nonzero vectors are perpendicular iff their dot product is zero. If we assert that the zero vector is perpendicular to everything, then this equivalence applies to all vectors, so the geometric statement of "perpendicular" coincides with the algebraic statement of $u \cdot v = 0$. This motivates the definition that two vectors in any inner product space are "orthogonal" iff $\langle u, v \rangle = 0$; in the case of Euclidean $\mathbb{R}^n$, orthogonal vectors are perpendicular vectors.

(This was originally an edit to Mohammad Riazi-Kermani's answer, but the edit was taken down.)
Answer: I've read that taking a dot product is just projecting one vector on the other, so a perpendicular vector will have no components in the other vectors direction.
Yes, that's the intuition.
But shouldn't this leave the length unchanged
Why? If you project the orthogonal vectors to each other, the length of the projected vector becomes zero.
Also "by definition", two non-zero vectors are said to be orthogonal when (if and only if) their dot product is zero.
This can also be demonstrated by Pythagorean theorem.
Comment: While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference.  Link-only answers can become invalid if the linked page changes. - [From Review](/review/low-quality-posts/1925452)
Comment: Hi @FD_bfa, I added essential parts of the answer to the body. Thank you for the suggestion.


email addr as _id instead of OID?

Question: I'm using MongoDB as a database back-end for a list serv I built and run for the two high schools in my district.
Each school will have its own database. Within the database for a school there will be collections.

There are multiple e-lists (subscription channels) at each school
Each subscriber can be subscribed to 1 or more e-lists (if they are subscribed to zero e-lists, they aren't a subscriber)
Each e-list (subscription channel) will have its own collection where each subscriber has a document (record).
The document for a subscriber within an e-list collection will have the email address of the subscriber and some administrative info

There will another collection containing all email addresses.

Each document in this collection will be a master record for the
subscriber.
Each doc will contain the info about the user (email addr, first
name, last name, etc.) and I will keep track of all the e-lists to
which they are subscribed so I don't have to iterate over all the
e-list collections to find which subscriptions a user has.

I know for certain that all email addresses will be unique. My question is this… should I:

Create the collections with a standard OID for the mongo _id field
and add an index for the email field
OR
Use the email addr as the mongo _id field since I know it will be
unique and forgo using OID

It might be worth noting that and I am running on a small footprint server.

Option 1 is more classic but the OID and email address are redundant
since the email address is unique.
From a storage perspective option 2 will be more efficient on disk
storage and possibly more efficient for RAM utilization
All my foreign key relationships are based on email addr. I can't
think of a reason I need the OID for the _id field.
Being a relative newbie with Mongo I'm wondering if I will be sorry
later on down the road if I don't go with option 1 (classic OID). In other words,
am I "over optimizing"?
Answer: Your proposed approach of using email address as _id will run into problems when:

A user wants to change their email address (not easily possible when other documents reference this email address)
A user wants to have multiple email addresses

Using the generated identifiers for _id field is a smart default.
You should most definitely not bother even thinking about space optimization until your dataset can potentially exceed the available RAM that would be realistically justifiable for the project (i.e. if you are developing an app for a small business, and their customer base grows, they likely can afford a server with 16 or 32 GB RAM).
Comment: Thanks for your answer. It is helpful in designing my solution.

email addr as _id instead of OID?


Why games with low system requirements are lagging?

Question: I got a HP laptop and I dual-boot Windows 7 Ultimate 64-bit with Kali Linux 64-bit.I installed Minecraft and Counter Strike to see how they are running and both of them were lagging a lot.I couldn't understand why,because the requirements are met for sure and I installed Windows a week ago...My specifications are:
-processor: AMD quad-core; 1.5 GHz; cache L2 512 kb
-RAM Memory:4 Gb; DDR3; 1600 MHz;
-Hard disk:500 Gb; 5400 rpm; SATA
-Video Card:integrated; AMD Radeon Mobility; HD 8330; DirectX 11 support
Also,three months ago I got ubuntu 14.04 installed and Minecraft was running smoothly (most of the time),I don't understand why it lags so much now,so if anyone knows what's the problem,please tell me.Thanks!
Comment: Minecraft isn't exactly a "game with low system requirements". ;)
Answer: HP and Canonicle work together so UBUNTU probably has better, if not proprietary display drivers than KALI.
Your AMD Radeon Mobility; HD 8330 uses shared system memory.
If Minecraft is multi-threaded, it is only in the most recent snaphots.
For your Win 7:
Defrag with a performance defrag program.
Look into shutting down un-needed services (alacrity is a program that will do this).
Use the latest video-drivers.
See if latest Minecraft does use multi-thread so you can use all your cores.


Fundamental group contains $\mathbb{R}$ or $\mathbb{Q}$

Question: Is there any topological space whose fundamental group contains $\mathbb{Q}$ or $\mathbb{R}$? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with suitable modification, such as universal coefficient theorem). But for a fundamental group, I've never seen any space that has fundamental group $\mathbb{Q}$ or $\mathbb{R}$. I think it is impossible, but how to prove it? Thanks in advance.
Comment: Note that, *as a group*, $(\mathbb{R}, +)$ is just the direct sum of an uncountable number of copies of $(\mathbb{Q},+)$.
Answer: Any group (including $\mathbb{Q}$ and $\mathbb{R}$) can be realized as the fundamental group of some space - in fact, for every group $G$, there is a two-dimensional CW complex with fundamental group $G$. See Proposition 1.28 of Hatcher here.
Comment: (+1) For the record, an explicit not-too-hard-to-see construction of a space with fundamental group $\Bbb Q$ is the following homotopy colimit: take $\Bbb N$-many copies of $S^1 \times [0, 1]$, glue the front of the $k$-th copy to the end of the $(k+1)$-th copy by a degree $k$ map, for all $k$. So, e.g., running along the top of the $2$nd copy of the cylinder twice is homotopic to running along the front of the first cylinder once, i.e., that element represents $1/2$.
Answer: Endow $Q$ with the discrete topology $Q_d$  and consider the classifying space corresponding to $Q_d$. Same construction with $R$.
Comment: If the OP knew what a classifying space was they wouldn't have asked this question.
Comment: so i hope, he will learn it,by the way if op new the aswer of his question he would not have asked it


Can a generator give me shock?

Question: Can an isolated AC system like a generator which is not system grounded will be able to give me a shock if I touch it one wire? I guess it should not, because then there would not be a path back to the source (the generator). But I highly doubt it and I was proved wrong in case of an inverter whose none of output wire were tied to ground still was able to glow a neon test screwdriver...
But an isolated DC system (like a battery) will behave as I had thought. No current would flow through my body to ground if I touch it...
Comment: "No current would not flow..." so it would or it would not?
Answer: There is a nice example: Tucker Phone. https://en.m.wikipedia.org/wiki/Tucker_Telephone
When i was in the military, tried it on myself, it's quite painful.
Of course, you have to hold two wires with two hands.
Comment: Upon my observation very small amount of current still flows across my body which means some voltage is there. It is because of capacitive coupling. Can anyone explain it to me?
Comment: "Of course, you have to hold two wires with two hands." means that your story is not relevant to the situation of the question.
Answer: Short answer is yes.
I once had an "isolated" generator that was not system grounded. I thought I was safe from shock. Turns out, you can get quite a healthy shock.
Isolated in terms of a generator generally means the neutral is floating, not that it is 100% isolated from ground. To prove it to yourself, take a multi-meter, and measure the voltage from the "isolated" neutral to earth ground. My generator measures about 60V, and gives you quite the tingle if you touch it!
Longer story:
I was working on a branch circuit on my cabin, the breaker was "off" but my isolated generator was running to power the other circuits. I touched the neutral wire and received a very healthy shock. I touched it again out of disbelief, and shocked my self again. It was when I measured the neutral wire with my DMM, I realized that my isolated generator was not as isolated as it should be.
Comment: What do you mean by healthy shock? Does it mean lethal as mains shock or very mild like a slight tingle.
Comment: Did you set your Multi-meter to Volts AC? 0 V is a suspicious reading. Do you get 240V when you connect to both the output wires? What were you using for a ground reference? As for the neon test screwdriver, it works using your body as a ground reference. Is your screwdriver the isolated tip or non-isolated tip? And as for the shock, voltage as low as 30 V can be lethal.  All it takes is a small current running through your heart to cause it fibrillate. When I work on mains power, i use the one hand rule. My other hand stays in my pocket!
Comment: I tried an experiment with my  inverter which outputs 240V AC which is an example of ungrounded system.I measured the  voltage of both output wires wrt ground and I found 0 V. Is it safe now to touch wires (not intending to do so) . But my neon test screwdriver still glows very dimly  I guess if voltmeter can't show reading the current shouldn't be high enough to shock me..
Comment: I used my water pipeline as ground reference and my Voltmeter was set to ACV and yes I get 240V connecting both wires. The screwdriver glows very dimly (you have to look into it to notice) but yes i got slight tingle when I touched the wires.
Answer: An AC generator is not the same as an DC battery.
If you have an unconnected thousand volt battery, you can safely touch one of the terminals. Because it is DC.
Not both obviously, since then you'd have an electrical path.
However, with AC, the electrical path will include capacitors.
For a generator for example, all the winding in the grounded housing will have a parasitic capacitance to ground. So will you, the large body of water walking on the ground.

Can an isolated AC system like a generator which is not system grounded will be ble to give me a shock if I touch it one wire?

Yes, if there is enough leakage current.
Which is why a Tesla coil works with fluorescent lights, if you hold them.

Can a generator give me shock?


What chords should I use in an aggressive orchestral track?

Question: I am trying to learn how to make epic (trailer) music. When I write melodic songs I use mostly major and minor chords and they work great, they give me exactly the sound I want, however those types of chords sound horrible when I use them in aggressive (staccato) epic tracks, so what kind of chords do you think are well suited for such tracks?
Comment: a peek at the score for "Mars, the Bringer of War" by Gustav Holst might be worth the effort
Comment: ...Especially when "Mars" manages to make major chord spam sound aggressive, too.
Comment: tbh, the entirety of Holt's Planet Suite sounds like the inspiration for a considerable number of popular film scores... [cough... John Williams] ;)
Answer: I think this might be a bit broad, so I'll set up a starter & let other people add their own ideas...
OK, take a simple minor chord, Em.
Voice it E, G, B & play it staccato, repeatedly.
Stodgy as all heck.
Revoice that as G, E - with a big hole in the middle, then drop the B an octave.
Whole different thing, but it's still Em.
Try again, drop that B bass to G below; change your RH voicing to E & B above.
See how much movement you gained & we haven't even changed chord or gone over 3 voices yet.
Comment: True, there's nothing wrong with using simple majors and minors as well as others, it's the voicing and instrumentation, as well as clever timing that do the job.+1.


Recessed lights and ceiling fan

Question: Replacing a ceiling fan with a fan with no light.  Want to add 4 recessed light cans.  The new fan only works with remote so no switch needed.  Can I wire the recessed cans to the same fan wire (2 wire romex)?  Can I just connect the black with black and white with white and add a dimmer switch for the recessed cans?
Comment: Depends a great deal on how this is wired. Does the power drop into the fixture box or into the switch box? Was the fan and light solely controlled by the remote, or is there 3-line wire (red, white and black wires) from the fixture to the switch?
Comment: Can you post photos of the insides of the fan and switch boxes?
Answer: No... & Yes... ?
perhaps you need to think this through a bit because ultimately you know (or will know) the wiring better than us.
In USA... Black "should be" HOT. White "should be" NEUTRAL. There should be a ground, you should use it. With that out of the way (noting that these are not always straight forward facts)...
Where does your supply come in? How do you intend to switch the lights on and off? If the supply goes through a switch, and your fan is currently wired to it, what do you suspect is going to happen when you switch that switch for the lights? (the fan will go off).
If the FAN has a remote, then it doesn't necessarily need a wall switch (but personally, I'd keep one anyways), however the lights will need a switch too... Right? You would need to add a switch for the lights. You can re-purpose the existing switch, but you'd still have to run a always HOT/NEUTRAL/GROUND to the fan (from a supply).
You have not mentioned how you intend to control the lights. If the supply side goes to the fan, then to the switch and back to the fan, then you can add lights off of the fan box provided you do not overload the box, or the circuit.  If the supply goes to the switch first, as is often the case, then you need to run another feed out of the switch box, to the fan, or the lights.

Recessed lights and ceiling fan


Is the vial necessary to Summon Greater Demon?

Question: Summon Greater Demon has material component "a vial of blood from a humanoid killed within the past 24 hours". I'm having trouble parsing whether "vial" is an actual vial or just a measure of amount? So if I've killed a guy, could I use that person's blood to cast Summon Greater Demon without a vial or do I specifically need to put his blood into a vial before the summoning can take place? Assume for a second that focus is not an option.
Answer: A vial is (probably) not needed but is a good idea
It's a fair question
On the one hand, "a vial of blood" could mean you need a literal vial.  On the other hand, perhaps you need a 'vial's worth of blood', since a vial is a unit of measurement.  A vial is, in fact, on the PHB list of Container Capacity as officially holding four ounces of liquid.  So it is a fair question as to whether you need specifically 'blood inside a vial', or whether 'exactly four ounces of blood in any sort of container' will do.  Given the ambiguity of meaning in 'a vial of blood', we won't be able to settle this by looking at the spell description itself, but we can look at other spells and see whether any patterns emerge.
Spells very rarely specify exact measures
A lot of spells have sympathetic magic material components in which a small amount of a symbolic substance is used.  For the dozens of spells that measure their components in "pinches", "bits", "pieces" (or "small pieces"), "handfuls", and "drops", I was able to find only one that unambiguously calls for a specific measured amount.  Clone requires "at least one cubic inch of flesh".  It would thus be very unusual for Summon Greater Demon to require exactly four ounces of blood.  However, there are a number of other spells that are similar to Summon Greater Demon in that they specify a liquid substance in a container that could be literal or could be a measurement.  "A vial" is also used in Commune, Summon Lesser Demons, Rime's Binding Ice, and Create Magen (and "a cup" is used in Armor of Agathys).  We can't do anything but circular reasoning here; either they all indicate a specific measurement, or they all denote a specific container type, but without knowing which we can't infer anything about the spell in question.
Spells sometimes call for specific liquid component containers
Some spells do actually specify that their liquid material component needs a specific container, not just a certain amount.  Hypnotic pattern, for example, requires "a crystal vial filled with phosphorescent material".  By saying 'a vial filled with material', we know that the vial itself, not just the measurement, is essential.  Similarly, Summon Elemental requires "water inside a gold-inlaid vial", and Create Undead requires "one clay pot filled with grave dirt [and] one clay pot filled with brackish water...for each corpse" (and note that by definition, an iron pot holds 1 gallon - but we don't know how much a clay pot holds).  Further, in each of these cases the material from which the container is made is specified (crystal, gold-inlayed, clay), while in Summon Greater Demons it is not.
A note on material components
We should remember also the rule that material components may be completely substituted by an component pouch or an arcane focus so long as the components don't have a specified cost.  Thus for Summon Elemental the vial itself really is an integral, noninterchangeable part of the spell, whereas for Summon Greater Demon both the vial and the blood can be substituted by other things.  It is reasonable to assume that a component pouch contains spare vials, less so that it has a constant supply of fresh humanoid blood, while a wooden wand has neither.  However, as Adeptus points out, the actual blood is necessary to draw the protective circle, which is an optional part of the casting.
A vial is likely not required, but preferred
We know that if it was important to specify the container as part of the spell, this could be done.  As in Hypnotic Pattern, not only could the spell say 'blood inside a vial' (rather than 'a vial of blood'), but it could tell us what material the vial would need to made from.  We know that simply saying 'a vial of blood worth 1gp' would mean the component could not be substituted by a focus or pouch.  We also know that if it was important to specify the exact amount of blood to be used, this could be done as well, as it is in the Clone spell.  Since none of these details are specified, Summon Greater Demons appears to be like the vast majority of spells, whose material components are no more detailed than those of Insect Plague: "a few grains of sugar, some kernels of grain, and a smear of fat."  In this spirit, it seems likely that all you need for the spell, absent a focus, is around four ounces of blood in some sort of container.
However, even if you have a focus to take the place of the material components, in this case casting with the components may be strongly preferred so that you can obtain the full benefit of the optional protective circle: "As part of casting the spell, you can form a circle on the ground with the blood used as a material component."  The spell may be telling us that four ounces of blood is the minimum needed to make a circle around "your space", such that with less blood you might be able to complete the spell, but not the circle.  Similarly, a narrow-mouthed vial, with which one can carefully control the pour rate of the blood, is the ideal container to efficiently trace the circle - were the blood held in a container like a bowl, you might need much more to complete the circle without leaving any gaps.
Comment: You need enough blood to pour out a circle of about 5' diameter: *As part of casting the spell, you can form a circle on the ground with the blood used as a material component. The circle is large enough to encompass your space.* I wonder if anyone's done the calculations on required blood per inch to form an unbroken line...
Comment: @Adeptus Upon closer reading, you don't actually need that much blood, since making the protective circle is an optional part of the spell: "...you _can_ form a circle on the ground".  So you need the full four ounces to make the circle, but you don't have to make the circle to cast the spell.
Comment: @Adeptus Four fluid ounces is just a hair over 7.23 in³. If we assume a uniform line width of 0.25 inches with the outer edge being exactly five feet in diameter that gives approximately 47 in² to cover for just the outer circle (not counting any internal designs that would normally be assumed for a magic circle), which those four ounces could cover to a depth of roughly 0.15 in. I have no concrete data, but my gut instinct is that that is probably more than good enough unless the surface is porous or wet.
Answer: A vial is a container, not a measurement
RAW you need the container, not just the blood. The spell says "vial of blood" - it means both the vial and the blood.
However, since the "vial of blood from a humanoid killed within the past 24 hours" has no cost, RAW, there is always one in your Component Pouch. How does it get there? RAW doesn't say.
Equally, RAW, it can be replaced with a focus.
Answer: I am going to drink a glass full of water
Despite that, I am not going to melt the glass into a molten pile and consume it as well.  I'm just going to drink the contents.
In English, referring to "a container of substance" as short hand for needing "the substance contained within the container" is standard, colloquial usage.  It is ambiguous, because language is ambiguous in most uses, but ambiguity doesn't make either reading invalid.
Read it either way
If you make a habit out of not having a component pouch and casting this spell, talk to your DM about "going to the source" for it.
If it is a one-off situation, talk to your DM when doing it.  It is possible that the PC doesn't even know, as every time they cast this spell (and the spell's instructions) tells them to have it in a vial!
Component Pouch
While it is perfectly reasonable to exclude "blood of a humanoid killed within the last 24 hours" as being a standard ingredient of a component pouch (I mean, maybe when you bought it it comes with that, but it seems extreme to include a delivery service that keeps it fresh and up to date), the vial itself has no cost and seems like something you might have in the pouch.
So even if focus/blood itself is ruled out, saying that the component pouch has a vial you can quickly fill is plausible and reasonable.
Answer: A strict reading implies you need a vial full of blood.
The list of items in Other Adventuring Gear section in the Basic Rules includes also a vial, with the relative cost. Hence, this implies that the container is actually one of the material components needed for the casting, the other being the blood from the killed humanoid.
Moreover, the "vial" entry in the Container Capacity section depicts the capacity of it, it is not a unit of measurement. It provides anyway information on how much blood you need for the summoning.
The DM may adapt the rules to the case at hand.
The DM may rule that you can collect the blood in a different container than a vial (a bottle, a flask, whatever may contain a liquid), taking into account that the required volume of blood is 4 liquid ounces. They may modify the quantity of required blood, too.
Moreover, they may rule that you don't need any container at all: for example, in the middle of a battle, your party kills an (humanoid) enemy, then you cast Summon Greater Demon and the blood streams out from the enemy corpse creating a crimson portal from which a Barglura comes out.
A really strict reading of the rules suggests that you have to be versatile in crafting new items...
A direct reading of the component requirements tells us that you need a vial made of blood.
Comment: You missed a permutation - perhaps you need to find a humanoid who is for some reason carrying a vial of blood, kill it, and take the vial.  That is, it is the vial that is "from a humanoid", not the blood itself.
Comment: @Kirt That's true: or maybe you have to find a corpse of an humanoid, killed within the last 24 hours (not necessarily by you), which was carrying a vial made of blood... interpretation running wild!


Are RNNs inherently flawed? Supervised Learning assumes IID data but sequential data is not IID

Question: From what I understand, Supervised Learning operates under the assumption that the data is I.I.D. It seems to me that the training procedure for RNNs is flawed. We receive observations in a sequential format -- this condradicts the assumption of IID -- that is, observations are temporally dependent on one another. Therefore, it seems to me that RNNs are flawed learning procedures. Am  I correct in my reasoning?
Answer: Your premise is wrong, it is not true that every (supervised or not) machine learning model assumes that the data consists or independent and identically distributed observations. If it was true, we wouldn't be able to model things like time-series, where the subsequent observations depend on each other, or other cases where the observations are correlated. For such cases we design models that assume dependence and take it into consideration, e.g. the time-series models, or recurrent neural networks. Moreover, you don't need RNN's to find dependence assumption in neural network architectures: in convolutional neural networks you use kernels that share weights and slide through the picture (or something else), so basically you assume that the collection of pixels within the kernel forms some kind of pattern i.e. depends on each other.
You may also find the following thread interesting: On the importance of the i.i.d. assumption in statistical learning
Comment: But I ran into a post by a researcher that said that reported that RNNs were failing for his problem because it assumed the data was IID.
Comment: @coolster1278 if you can provide link or exact quote, I could comment on this, but otherwise it would be a guessing game.


My dog who is other wise very gentle has started hunting and killing and eating baby rabbits!! How can i stop him?

Question: My dog is one of the gentlest dogs I have ever met. Up until a couple days ago I wouldn't think he could hurt a fly. That all changed about 3 days ago when I heard a animal scream and I ran to see what it was and all I seen was two tiny feet hanging out my dogs mouth right as he swallowed it. I yelled at him and he ran and snatched something up a few feet away and I yelled for him to drop it but instead he ran off and before I could get to him he had swallowed it. I still had no idea what it was until he once again went and snatched up something off the ground. By this time my husband heard me yelling and had come to see what was going on so I told him my dog was eating another animal and together we cornered him and he dropped what he had and to my shock it was a tiny baby rabbit!! So small it fit in my hand!! I scolded him put him inside and didn't let him out for the test of the night. The next day I let him out. To my surprise he caught yet another one from a different nest and was about to eat it when I yelled at him and he dropped it. So I know he knows better. I punished him again by not letting him go out on his own and putting him on a leash when he did. I had hoped he understood but this morning I let him out to use the bathroom but instead he went straight back to the spot he dropped the baby rabbit and went to looking for it. So I grabbed his collar and scolded him and put him back inside. I don't want to see my dog as a baby killer!! I love my dog and this is killing me to have to fear he is eating these poor tiny rabbits and pretty much swallowing them whole!! Is there anything I can do? I can't just take away his freedom and keep him locked up or tied up. What if he starts hurting other animals? Will he kill any kittens or turn vicious if I stop him from being able to run free? I am just so disappointed in his behavior I don't know what to do. How can a dog that is so gentle he won't even take food from your hand end up being a baby rabbit killer? I know I can't stop him from finding them bc he has a hell of a nose on him and can smell out their burrow so how can I stop this from happening? Please someone help!! Will he become obsessed and kill them all? Ismthere any disease baby rabbits can pass to him from him eating them pretty much whole? I am desperate not to feel this way towards him. ☹
Comment: Welcome to pets.SE! Please give a focused question, or a detailed aim you want to reach. Because dogs are hunting animals, predators, it may not be possible to aim for him not hunting if he is unsupervised. If we have a clear question/aim we can better help you answer/reach what you want.
Comment: Do you know what breed your dog is? It seems he has a strong prey drive. It is surely very unlikely he'll turn vicious towards anything he's been perfectly well behaved around, but you should assume any similarly small animals, including kittens, are also in danger from him.
Answer: Maybe the place to go with this is to understand nature rather than trying to change it. Your dog is a predator. His ancestors, for a very long time indeed, did exactly this sort of thing. And they did it often enough not to starve, long enough to raise pups of their own.
If you are going to have a dog, it is kind of difficult to be upset with him when he does dog things.  Now that he knows what rabbits are and what they taste like, it's going to be tough going to stop him trying to get more. Imagine stopping yourself from having cake or ice cream.
Maybe the best you can hope for is to divert him and reduce the rate of his activity. Keep him well fed, especially just before walkies. Keep him on leash whenever there might be a rabbit about. Train him to obey command such as "Stay!" and "Sit!" and "Come!" and "Heal!" (You need to work at these.)
As to your dog's health: a dog's digestive system is adapted to this. If a rabbit is the worst thing you ever see him eat, consider yourself not too badly off. It is unlikely to harm him.
I hung out at my brother's place last summer. He has a dog (female German Shepherd about 3 years old) and two cats. One cat is so old she has completely lost litter tray discipline, and frequently poops in the middle of the dining room floor. Beloved cat, and this is simply tolerated. She's extremely old, possibly as old as 20, so it can't continue much longer. So, one morning, I got up before everybody else. Pet the dog good morning. Noticed the cat had left a chocolate bar on the floor, and went to get paper towels to clean it up. Turn back, and the poop is gone. Dog sitting there licking her lips.
OK, doggy. You don't get to lick my face any more.


Cyclic tour minimizing total weight

Question: I asked the following question on math.se but it wasn't really answered so moved it over here as I feel it's more relevant.
I saw the question below on an old stack exchange question when looking to understand NP-completeness. At first, I thought I understood it so I gave it a go but ended up more confused than before (I'm a maths student and computational complexity is my first foray into CS and I'm struggling to get to grips with it).
Question is as below:

Given as inputs integers $a, b$ and $c(i,j)$ for each $i,j \leq a$, decide if there is a permutation $\tau$ of $\{1,2,\dots,a\}$ such that
$$
c(\tau(a),\tau(1))+\sum_{i=1}^{a-1}c((\tau(i),\tau(i+1))\leq b
$$
Prove this problem is NP-complete. Hint: find a reduction from the Hamiltonian circuit decision problem (given a graph, decide whether it contains a Hamiltonian circuit).

It seems very similar to the decision version of TSP (where given a set of cities with the distances between each pair of cities, and a length $k$, does there exist a circuit connecting all cities of length less than $k$).
I've read the proof for NP-completeness of TSP and I feel like I understand the reduction from Hamiltonian circuit decision problem to TSP, but I can't get it to translate into a reduction from HC to this problem.
My understanding to compare this to TSP, is that $a$ is the number of cities, each $c(i,j)$ represents the distance between city $i$ and $j$.
My (fairly poor) attempt at this reduction is as follows;
The problem is clearly in NP as we can non-deterministically "guess" a permutation and calculate the given sum and verify that it's at most $b$ in polynomial time.
Then, given a graph $G=\langle V,E\rangle$ where there are $a$ vertices, say $1,2,\ldots,a$, and $b$ edges of length $c(i,j)$, where $i \neq j$ and $i,j\leq a$, construct the completion of $G'$ of $G$ and a weight function that assigns any edge of $G'$ a 1 if the edge is present in $G$ and 2 if not. This can be computed in polynomial time.
Then if $G$ contains a Hamiltonian circuit (consisting of $b$ edges), this path has weight at most $b$ in $G'$ and so the sum of the lengths of this path is at most $b$ and so the permutation corresponding to the order of this path satisfies the original decision problem for these numbers $a,b$ and $c(i,j)$.
Conversely, if given the numbers $a,b$ and $c(i,j)$ and a permutation $\tau$ that satisfy the problem the associated graph G contains a path starting at $\tau(a)$ and finishing at $\tau(b)$. Thus G has a Hamiltonian circuit.
Thus HC reduces to this problem and hence it is NP-hard and so NP-complete.
I can tell it's wrong but I'm not sure how to properly describe a function relating to Hamiltonian circuits and lengths into weights in $G'$.
Any advice and or pointers would be much appreciated.
Answer: The problem you describe is exactly asymmetric TSP. As your reduction shows, asymmetric TSP generalizes the Hamiltonian circuit problem, which is known to be NP-hard.
Usually we consider metric versions of TSP, in which the triangle inequality $c(i,k) \leq c(i,j) + c(j,k)$ holds, and often the TSP instance is symmetric, $c(i,j) = c(j,i)$. Your reduction shows that (symmetric) metric TSP is also NP-complete, since the cost function you construct is symmetric and satisfies the triangle inequality.

Cyclic tour minimizing total weight


What is the output of a 2-tag system?

Question: Considering a 2-tag system as defined by wikipedia, how do I identify the result  of the computation?
Answer: There are two types of Post tag systems: with halting symbols, and without.
A Post tag system with a halting symbol gives its output in the same way that a standard Turing machine does: on the tape. Whatever is written on the tape when it hits the halting symbol is its output.
A Post tag system without a halting symbol is a bit trickier. It only halts when the tape has too few symbols left to do another step of processing. So there's no way to give any output on the tape. Thus, its output is the sequence of intermediate states the system has been in in the process of producing the result.
The Wikipedia article you linked gives an example of this: a halting-symbol-less 2-tag system to compute the modified Collatz sequence.

What is the output of a 2-tag system?


Why is Vipassana centred around the breath, when it's meant to be a state of observing sensations?

Question: I can't seem to reconcile the two ideas that Vipassana meditation involves both concentrating completely on the breath while at the same time observing bodily sensations. Aren't these two mutually exclusive? How can one concentrate on the breath while being open to new sensations?
Answer: When you watch your breath, seeing the quality or character of each in-breath and out-breath allows you to really see your feelings (not sensations, that's a bad translation). Breath is the best pointer at your emotional state.
Answer: Vipassana does not involve concentrating solely on the breath. Instead (one form of) it involves resting your attention on the breath. While keeping your attention on your breath your awareness naturally has to be open to unconsciously "decide" whether or not the attention should change to something else, so already it is naturally open in this situation. By placing your attention on your chosen object of meditation you are implicitly opening your awareness.
At first it's not obvious: you may feel that you are really forcing your attention onto your breath at the expense of other sensations, but as you relax and open further you will begin to perceive your awareness with much more clarity.
Source: The Mind Illuminated
Answer:
Aren't these two mutually exclusive?

No. When the breath is known within the body, simultaneously sensations within the body will be felt. Also, the impermanence of the breath will be experienced.
Answer: If you simply observe everything, eventually, the mind starts wandering.
So, observation of the breath in vipassana serves a few functions:

It anchors the mind into the task of observing the breath and you need mindfulness to keep returning to this task, to prevent the mind from wandering.
It calms the mind down.
With the breath, there comes other physical sensations that are observable, like the movement of the abdomen, movement of air through the nasal passage etc.
With the breath, there are related mental states that can be observed like calming down, agitated, unhappy, happy etc. because negative mental states tend to cause heavy agitated breath and positive mental states have lighter breath.
The relationship or mutual influence between mind and body can be observed.


WQL Subquery as field value, CIMV2 WMI WQL query for WMI-Filter

Question: I need help on a WMI query for a Group Policy WMI-Filter.  I'm querying the win32_group namespace within the root\CIMV2 WMI provider.
I am doing a search for a Local Security group (like "Administrators") on several window's computer (XP-8, server 03-12).  When the query runs on a computer I would like it to inject the computer name into the equivalency field for the Domain.
The base WQL statement looks like this:
SELECT * FROM win32_group
WHERE Domain="currentComputerName" and Name="Administrators"

My goal is to run a WQL statement along these lines, but it is a bad WQL statement:
SELECT * FROM win32_group
WHERE Domain=(SELECT Name FROM Win32_ComputerSystem) and Name="Administrators"

It is important that I am able to inject the current computer's name for the Domain.  These computers are part of a Active Directory (AD) domain. By default the win32_group will search all of the AD Domain's groups in addition to the Local Computer's security group (if omitting the Domain parameter).  By specifying the local machine name as the domain the query's performance is multitudes faster - which is essential for its purpose.
I have tried some query variations in WBEMTEST with no luck. Does anybody have any ideas on how to inject the computer's name into the WQL WMI Query?  Is there possibly a constant that I could use, like HOSTNAME at the command prompt?
Thank you for your help!
Answer: What language are you writing this in? Or to be clear, what programming or scripting language you are using to run these WMI queries?
In PowerShell, you could do this by running the following code:
Get-WMIObject -Query "SELECT * FROM Win32_Group WHERE Domain='$env:USERDOMAIN' AND Name='Administrators'"


Is there a bacteria that eats/reduces DMS?

Question: I have done a few kettle sours and in some cases, a wild yeast or heterofermentative bacteria ferments out quite a bit of the sugars. In the past, I've used DME to bring the gravity back up for the boil, but it would be good to have the option of just skipping the boil, however, there's way too much DMS in the wort at this point.
Answer: Yes there are sulphate-reducing bacteria. These are mostly found in marine biology. I don't know of any of these bacterium useful in brewing though (maybe just ignorant).
In your case DME will have low SMM and DMS, because it has already been mostly boiled out in proccesing. Though I would still do a short boil 5min boil and chill for sanitation.
Fun Facts
Some yeast can convert DMSO into DMS.
DMS can be scrubbed with cO2.
DMS can be perseived with at very low levels.
DMS at high levels can smell and taste like shrimp / seafood water.
DMS has half life of 40 minutes at it's boil temp (99 21°F) so a 90 minute boil or wort usually is enough to get the levels below the detection threshold of most palets.
Once DMS turns into DMSO it can't be boiled off. Its boil temp is > than waters.
Comment: I'm not concerned about the DME, rather the existing wort has noticeable levels of DMS from the mash. It also has some alcohol content. If I boil it, I'll need to use the DME to bring the gravity back up to ferment.
Comment: @Wyrmwood well if you boil the beer up the alcohol will be gone. But I was referring to boiling a DME solution and chilling, then adding that to your beer.
Comment: Yeah, boiling just the DME doesn't do anything to reduce existing DMS levels though. I've either got to find an alternative to boiling to reduce DMS or add DME since the sugars are now much lower and fermentation will essentially be restarted.


How to get my keyboard backlight back after Mountain Lion upgrade?

Question: I have a non-Unibody MacBook Pro running 10.8.2. After I installed Mountain Lion, the keyboard backlight does not light up any more. Is it a connection with the new OS, or is it a defect of my Mac?
Comment: What happens when you press F5 or F6? You may need to hold Fn (function) when doing so depending on your keyboard settings. I want you to look for [this overlay](http://i.stack.imgur.com/a4Kxl.png) to appear on your screen when pressing F5/F6, or Fn F5/F6.
Answer: You're not alone, expect an OS X update to address this very soon, until then others have had success by disabling the auto dim in low light feature.
System Preferences > Keyboard > uncheck "Adjust keyboard brightness in low light".


Is Subsonic 3.0 safe for production use?

Question: I'm interested in using Subsonic 3.0 for developing a new ASP.Net MVC application that will be going into production use soon. Would this be a feasible option? Does anyone have an idea when 3.0 is going to be released? Is anyone else using it on a production MVC application?
Answer: I'm getting there - we're in the clean up phase and I'm happy to say Eric's jumped back into the fray :). I ran perf tests the other day and fixed up some bottle-necks and the core is pretty much where we want it to be.
I have no idea when it will be ready...
Comment: Thanks Rob, I've decided to just start using it and see how far I get. I'm hacking on the source as well, if I accidentally fix something I'll send in a patch :)
Answer: I'm not using it on a production site yet, although I am building a future production site with it, but its a hobby project so I can't get fired if the DAL has a bug and SS4 will probably be out before my hobby project.
So far I have not run across any show stoppers or actually any bugs at all and in the past I've found that if you do find an issue Rob is pretty good about getting a patch in pretty quick.  So I guess to answer your question it's probably not "production ready", but in playing around with it I felt it was pretty darn solid and chose it over LINQtoSQL or Entity Framework.  I'd recommend isolating your DAL (IRepository) that way if you run into something with SS you can swap out with something else fairly quickly.
Regarding as to when it is ready - I'll let Rob chime in on that one.  I think he is using it for building out some other app so he's sort of using that to dog-fooding SS3 - my guess is it will be a couple months yet though.
Answer: I think for me the only show stopper is the fact that it doesn't run in medium trust.
While for nearly all projects we work on this isnt an issue as its a dedicated environment for fast "get-em up" sites that subsonic is REALLY good for this sucks as it means that a shared host is not possible... :(
but subsonic 2.0 is great for that even if its lacking linq support.
p.s i know they are working on this :)


How do I spread taint to a certain area in Thaumcraft?

Question: I'm trying to make taint spread to a certain area in my world, as in, to a small NPC structure. Is there any way I can do this? I even tried using MCEdit to set the biome, but there's no "enter biome ID" thing, and Tainted Land isn't listed.
Comment: the wiki talks about spreading taint and creating taint http://thaumcrafttwo.wikia.com/wiki/Taint
Comment: As I remember, Thaumonomicon "bad science" stuff also has things to spread taint, for now I can remember only Liquid Death, but there is more stuff...
Answer: You can create tainted land by throwing "Bottled Taint"

This research is forbidden knowledge (minor). These nasty little vials allow you to create taint patches anywhere you like, by using (right-clicking) them on the ground.
  Source: http://thaumcraft-4.wikia.com/wiki/Bottled_Taint
Answer: From How Taint Spreads on Bayrattack's link:

A local infection will not penetrate the ground and only infects the blocks on the surface ... The infection also only spreads to the blocks next to other already infected blocks ...  If there are blocks the Taint cannot infect adjacent to infected blocks, the Taint will not be able to spread beyond them. As will be shown later, this can be used to create barriers that can temporarily stop a local infection from spreading. Taint will mostly infect naturally occurring blocks ... Taint will not infect most player-made materials

From this, you can sculpt a "path" of naturally occurring blocks, i.e. dirt, surrounded by man-made blocks, i.e. wooden planks, to the intended area.
Depending on your version of Thaumcraft, you can also research Concentrated Taint and drop it in the structure you want tainted.
Comment: Recommend mentioning that taint nodes, the source of taint, can also be moved to infect an area.  There are also other ways.
Answer: Moving a tainted node to the area should do it but it would take time to change the blocks and biome. And since your in creative you should also be able to place tainted blocks.


What is Encapsulation and how can it defend abstractions against corruption?

Question: It's quoted from a report by Bjarne:

Encapsulation – the ability to provide
  guarantees that an abstraction is used
  only according to its specification –
  is crucial to defend abstractions
  against corruption.

Can someone explain this?
Thanks
Answer: Let's say you have a class with public methods that you must use to perform some action.  The specification of the class say that, in order to do this action, you must configure the class in a specific way (call this method, set this property, etc).
The problem with situations like this is that it might not be clear what needs to happen or in what order.  So the API for the class is hard to use and confusing for the majority of developers.
With encapsulation, you can "encapsulate" not just the class but the algorithms to use it within a second class.  This second class sets up the original one, configures it, and manages its lifetime.  It allows you to access the API without needing to know how to use it correctly, as the encapsulating class takes care of that.  This is sometimes called the Facade pattern.
Your quote also says "is crucial to defend abstractions against corruption."  What this means is that when you abstract some process into a class, different implementations of that process should not require the abstraction to be handled differently.
For example, you might have two implementations of a report writer class.  You should be able to treat each of them exactly the same without ever knowing how they are implemented (the meaning of abstraction).  However, if one cannot be run in a multithreaded apartment state (MTA), you have to "know", before you use it, that it is time to transition to an STA thread.  This magical "knowing" is required by the implementation of the class.  This is a "leaky abstraction."
With encapsulation, you could prevent this "leak" by, within the encapsulating class, making the transition to an STA thread within the encapsulation, preventing the abstraction from leaking details of its implementation.
Answer: It means that the object grant premission only to certain things it needs to expose, and deny you from using data it doen't want you to use.
The most classic example is properties:
Yout fields will be private (or protected).
If you would like to expose them for read or write, you'll add a getter\setter, accordingly.


adding instances of dates repeating in column A across 11 tabs (sheets) in a sheet

Question: How would I create a formula that shows a sum tally of how many instances each date appears on column A across multiple sheets?
I want to make a formula that will display a total tally of how many orders I had each day. All of my orders are spread out out over multiple sheets. The 1st eleven sheets in my master sales sheet contain the entries for all my sale orders. Every sale is given a new row and on those 11 sheets the date of every order is entered in to Column A of those rows. The same date will appear multiple times since I have multiple sales each day. There are dates entered in other places on those sheets but they aren't referencing dates of sales they are referencing other things that aren't related. I only want to tally up the amount of time each date appears in column A across those 11 sheets and nowhere else.
Answer: The solution that follows is based fundamentally on the COUNTIF function to get a conditional count, and FLATTEN to combine the dates from multiple columns into a single column.   BYROW and LAMBDA are leveraged to create a single formula that can be applied across an entire range. Those 4 functions are listed immediately below however an effort was made in notes to describe the purpose of all functions used, and to link their names to Google support articles that can provide more detail.
Your question describes 11 columns spread across the same number of sheets.  This solution includes 3 comma-separated ranges (not 11).  It is a simple matter to add the additional ranges to the formulas below using the same syntax.
COUNTIF
The COUNTIF function returns a conditional count across a range.
Syntax: =COUNTIF(range, criterion)
FLATTEN
The FLATTEN function combines values from one or more ranges into a single column.
Syntax: =FLATTEN(range1, [range2, …])
BYROW
The BYROW function groups an array by rows by application of a LAMBDA function to each row
Syntax: =BYROW(array_or_range, LAMBDA)
LAMBDA
The LAMBDA function contains a set of names and a formula_expression that uses them. To calculate the formula_expression, you can call the returned function with as many values as the name declares.
Syntax: LAMBDA(name, formula_expression)
NAMED RANGES
I find using NAMED RANGES instead of <SheetName>!<cell>:<cell> addressing makes formulas easier to follow visually and often easier to maintain. You can do the same, using a similar or different naming convention, or replace the NAMED RANGES with actual range references, as you prefer.  Below are the names used for the 3 date columns used in the formulas.

Name
Range

_s1A
Sheet1!A:A

_s2A
Sheet2!A:A

_s3A
Sheet3!A:A

Formula 1
Get count beside existing dates
This formula addresses the use case where there is a column of dates in a sheet (like you showed in your image) and you want to count how many times each of those dates are repeated in a FLATTENED list of all the date columns throughout the spreadsheet.

BYROW  is used to apply the LAMBDA formula to all rows in A:A
LAMBDA function is used to apply the same formulas to each row. the arbitrary variable r is used to represent the current row being processed.
IF combined with COUNTA is used to test for, and skip empty cells
IF combined with DATEVALUE is used to skip non-dates that may be in the list, and since a DATEVALUE will error out on non-dates, it is wrapped in IFERROR to return a NULL value (blank) for non-dates
COUNTIF is now used to count the number of occurrences of r in our list of dates

=BYROW(A:A, LAMBDA(r,
   IF(COUNTA(r),
     IFERROR(
        IF(DATEVALUE(r),
           COUNTIF(FLATTEN(_s1A,_s2A,_s3A),r),
           FALSE)),
     IFERROR(1/0))
))

Formula 2
Get the list of dates
If, unlike Formula 1, the count will not be hosted beside an existing list of dates, a list of dates must then be generated.  This list is built by flattening all the date columns throughout the spreadsheet, then filtering out any duplicates, as well as non-date values.

LAMBDA function is used so that once the list of dates is flattened and sorted, it can be reused based on the arbitrary  variable d rather than having to write it out again (in the FILTER function for example)
FLATTEN function combines all the date columns into a single column and SORT is used (optional) to return the dates in ascending order
FILTER is used in combination with IF and DATEVALUE to remove non-dates, and since a DATEVALUE will error out on non-dates, it is wrapped in IFERROR to return a NULL value (blank) for non-dates
Wrapping the formula in UNIQUE allows us to remove duplicates as we only want to check the count for each date once

=UNIQUE((LAMBDA(d,
   FILTER(d, IFERROR(IF(DATEVALUE(d),TRUE),FALSE)))
   (SORT(FLATTEN(_s1A,_s2A,_s3A)))
))

Combined Formula
Formula 2 as input array for Formula 1
While you can theoretically keep Formula 1 separate from Formula 2, the two formulas can be combined.   Formula 2, whose result is an array, becomes a direct replacement for the range A:A in Formula 1's BYROW function.
=BYROW(
  ( UNIQUE((LAMBDA(d,
     FILTER(d, IFERROR(IF(DATEVALUE(d),TRUE),FALSE)))
     (SORT(FLATTEN(_s1A,_s2A,_s3A)))))
   ), LAMBDA(r,
     { r, COUNTIF(FLATTEN(_s1A, _s2A, _s3A), r) })
)

Images
Formula 1: Get count beside existing dates
non-dates (highlighted in red) are ignored: text, numbers, empty cells

Formula 2: Get the list of dates

Combined: Formula 2 as input array for Formula 1


Why Earth's atmosphere behaves like an ideal gas?

Question: In every book I found this sentence like an assumption, without explananions, somebody can help me understand it better?
Comment: ...because it is made up out of gas, under low-ish pressure?
Answer: The Earth's atmosphere is mostly nitrogen and oxygen, both of whose behaviors are very close to ideal at the temperatures and pressures found in the atmosphere. Nitrogen, the dominant gas in the atmosphere, comes particularly close to exhibiting ideal behavior. Gaseous oxygen exhibits about a 3% departure at 20 atmospheres at standard temperature, with departures from ideal reducing more or less linearly at reduced pressures.
There is one component whose behavior is markedly non-ideal, and that is H2O. Water vapor is a trace gas, at most a few percent of the atmosphere (and that extreme occurs only in very humid tropical regions). Unless you are studying cloud physics, you can pretty much ignore the departures from the ideal gas approximation. The errors that result from assuming ideal behavior are generally less than one percent.
Non-ideal behavior becomes important if you want to model the atmospheres of Venus or the gas giants because of the higher pressures. For example, the CO2 that is by far the dominant component of Venus's atmosphere is a supercritical fluid at the surface of Venus. Departures from ideal behavior can be quite extreme in this regime.
Comment: So can we consider air as ideal gas for any temperature ?
Comment: @magemaro - No.
Comment: Then are they considered ideal at temperatures below toom temperature?
Answer: Look at the definition of ideal gas .

An ideal gas is a theoretical gas composed of many randomly moving point particles that do not interact except when they collide elastically. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. One mole of an ideal gas has a volume of 22.7 L at STP (standard temperature and pressure).
At normal conditions such as standard temperature and pressure, most real gases behave qualitatively like an ideal gas. Many gases such as nitrogen, oxygen, hydrogen, noble gases, and some heavier gases like carbon dioxide can be treated like ideal gases within reasonable tolerances

From the list and knowing the composition of the atmosphere one can see that the simple classical model of non interacting elastically scattering molecules is good enough for  most purposes. Even though there exist interactions and scatterings follow quantum mechanical laws, to first order the ideal gas is a good approximation.
It is an assumption, but a good approximation of what is going on really.
Answer: Any gas behaves as an ideal gas under high temperature an low pressure. Atmospheric oxygen is at 25deg celcius which is greater than it's critical temperature which is -150 deg celcius. And the pressure is somewhere around 159 mm of Hg (21.1 kPa). Hence atmospheric gases is said to behave like that of an ideal gas.
Comment: Is that relevant? What matters is the number density of particles in the gas, not just the number density of oxygen molecules. And how do you go from this pressure to saying "hence" it's an ideal gas?
Comment: One atmosphere is 760mm of Mercury.
Comment: Partial pressure of atmospheric oxygen is roughly 159 to 160 mm of Hg.