QM with complex eigenvalues

Question: What class of theories/physical systems own finite/infinite complex eigenvalues? I do know that e.g., quasinormal modes of BH do have complex eigenvalues, but are they finite or infinite in number? What about Conformal Field Theories with complex eigenvalues? What are the general spectral properties of CFT? By the other hand, what kind of non-hermitian operators could provide spectra with infinite complex eigenvalues in pairs ( complex-conjugated)? What about the problems in the "new"/"recent" quantum graph theory?Is there some "infinite" graph with complex eigenvalues? Remark: I am not an expert on spectral graph theory but I do know a little bit about its methods.
Comment: There are a lot of questions in one post there.  I think it would benefit from reorganizing into a smaller number of questions.  If the gist is a more general "how are complex eigenvalues used in physics" it may get more response if phrased that way.
Answer: Complex-valued eigenvalues can be used to introduce the concept of electromagnetic mass and charge.
Below I will provide an example of the Lorentz invariant model that uses complex-valued eigenvalues as a key component, but nevertheless allows for well defined momentum density with real valued mass density square. The "complexity" of the eigenvalues in this model is used for defining the complex-valued charge density that can further be split into electric and magnetic charge densities.
Let's start with the free Dirac equation in spinorial form:
$\begin{array}{
cc}{{\partial{}}^{\mu{}\dot{\nu{}}} \ {\eta{}}_{\dot{\nu{}}}\ =-\ im{\xi{}}^{\mu{}}}\\ \\{{\partial{}}_{\mu{}\dot{\nu{}}} \ {\xi{}}^{\mu{}}=-\ im{\eta{}}_{\dot{\nu{}}}}\end{array} \ \ \ \ \ \ \ \ \ \ (i)$
where
$
\left({\partial{}}^{\mu{}\dot{\nu{}}}\right)=\ \left[\begin{array}{
cc}
{\partial{}}_0+{\partial{}}_3 & {\partial{}}_1-i{\partial{}}_2 \\
{\partial{}}_1+{i\partial{}}_2 & {\partial{}}_0-{\partial{}}_3
\end{array}\right]=\
{\partial{}}_0 +{\partial{}}_1{\sigma{}}_1 + {\partial{}}_2{\sigma{}}_2+{\partial{}}_3{\sigma{}}_3
$
$
\left({\partial{}}_{\mu{}\dot{\nu{}}}\right)=\ \left[\begin{array}{
cc}
{\partial{}}_0-{\partial{}}_3 & -{\partial{}}_1-i{\partial{}}_2 \\
-{\partial{}}_1+{i\partial{}}_2 & {\partial{}}_0+{\partial{}}_3
\end{array}\right]=\
{\partial{}}_0 - {\partial{}}_1{\sigma{}}_1^T - {\partial{}}_2{\sigma{}}_2^T - {\partial{}}_3{\sigma{}}_3^T
$
Now let's consider modification of the free Dirac equation $(i)$ by replacing constant "mass terms" with the variable electromagnetic field spinors:
$
\begin{array}{cc}{{\partial{}}^{\mu{}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}=\
+f_{\nu{}}^{\mu{}}{\xi{}}^{\nu{}}} \\ \\
{{\partial{}}_{\mu{}\dot{\nu{}}}{\xi{}}^{\mu{}}=\
-{\dot{f}}_{\dot{\nu{}}}^{\dot{\mu{}}}{\eta{}}_{\dot{\mu{}}}} \end{array} \ \ \ \ \ \ \ \ \ \ (ii)
$
Here second-rank spinors of electromagnetic field $ f_{\nu{}}^{\mu{}}$ are defined as follows:
$
f_{\nu{}}^{\mu{}}=\ \left[\begin{array}{
cc}
f_1^1 & f_2^1 \\
f_1^2 & f_2^2
\end{array}\right]=\ \left[\begin{array}{
cc}
F^3 & F^1-iF^2 \\
F^1+iF^2 & -F^3
\end{array}\right]=F^k{\sigma{}}_k,\ \ k=1,2,3
\
$
where  $F^k=E^k-iB^k $
and complex-conjugated spinor $\dot{f}^{\dot \mu}_{\dot \nu}$ is defined as
$\dot{f}^{\dot \mu}_{\dot \nu}=\bar{ f_{\nu}^{\mu}}$
Equation $(ii)$ will replicate the form of the free Dirac equation $(i)$ if we require that spinors $\xi$ and $\eta$ are eigenvectors of the electromagnetic field spinors $ { f_{\nu}^{\mu}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$:
$ \begin{array}{
ccc}
f_{\nu{}}^{\mu{}}{\xi{}}^{\nu{}}=\ \lambda{}\ {\xi{}}^{\mu{}} \\
\\
{\dot{f}}_{\dot{\nu{}}}^{\dot{\mu{}}}{\eta{}}_{\dot{\mu{}}}=\bar{\lambda{}}\
{\eta{}}_{\dot{\nu{}}}
\end{array} \ \ \ \ \ \ \ \ \ \ \ \ (iii) $
Indeed, by applying $(iii)$ in $(ii)$ we obtain
$ \begin{array}{cc}{\partial{}}^{\mu{}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}=+\ \lambda{}\
{\xi{}}^{\mu{}}\\ \\{\partial{}}_{\mu{}\dot{\nu{}}}{\xi{}}^{\mu{}}=\
-\bar{\lambda{}}\ {\eta{}}_{\dot{\nu{}}} \end{array} \ \ \ \ \ \ \ \ \ \ \ \ (iv)$
Eigenvalues $\lambda$ and $\bar\lambda$ in $(iii)$ and $(iv)$ are the well known invariants of the electromagnetic field:
$ \begin{array}{
ccc}
{\lambda{}}_{\pm{}}=\ \pm{}\sqrt{E^2-B^2-2i\bf{EB}} \\
\\
{\overline{\lambda{}}}_{\pm{}}=\ \pm{}\sqrt{E^2-B^2+2\bf{EB}}
\end{array} $
The momentum density $P_\mu$ can now be constructed by usual way as a sum of two isotropic 4-vectors $p_\mu$ and $\hat p^\mu$:
$P_{\mu{}}=\ p_{\mu{}}+\ g_{\mu\nu}\hat{p}^\nu$
where
$\begin{array}{cc}
p_{\mu{}}=\ \frac{1}{2}\ \left({\xi{}}^+{\sigma{}}_{\mu{}}\xi{}\right) \\ \\
{\hat{p}}^{\mu{}}=\
\frac{1}{2}\left({\dot{\eta{}}}^+{\acute{\sigma{}}}^{\mu{}}\dot{\eta{}}\right)\end{array}$
In spite of the complex values of the "mass densities" $\lambda$ and $\bar\lambda$, the momentum densities of the matter fields $P_\mu$ are:

real valued
satisfy the continuity equation $
   {\partial{}}_{\mu{}}P^{\mu{}}=0$ as a consequence of equation $(ii)$
"mass density square" equals to
$P^{\mu{}}P_{\mu{}}=4{\left\vert{}\lambda{}\right\vert{}}^2$ and
hence is always real and positively defined
World vector $P_\mu$  is
an eigenvector of the stress-energy tensor of the electromagnetic
field

It is worth noting that in this model the momentum density vector $P_\mu$ is always time-like, and its time-like component $P_0$ is always positive, hence no solutions with negative energies are allowed.
So we have shown that using complex eigenvalues $\lambda$ and $\bar\lambda$ the mass density can be expressed via electromagnetic field strengths $\bf{E}$ and $\bf{B}$.
Now we can develop the concept of the charge density that can also be expressed via $\bf{E}$ and $\bf{B}$.
We can do it by requiring that matter field equations $(ii)$ are equivalent (or reduced) to Maxwell equations. Physically that means that particle's own electromagnetic field evolution is dynamically balanced with evolution of its source – particle's spinorial field, so that the total field configuration remains stable in time.
As shown here, in the case of transverse plane waves the Maxwell and matter field equations become equivalent if the following relationship between the mass and charge densities is satisfied:
$ J^{\mu{}}=\bar{\lambda{}}\ P^{\mu{}} $
From this we conclude that, in the case of the transverse plane waves, electromagnetic field invariant $\bar\lambda$ plays the role of the charge density (while $\lambda$ plays the same role for anti-particles). Generally $\bar\lambda$ is complex valued, hence allowing for both non-zero electric and magnetic charge densities.
It is also demonstrated that in the case of the transverse plane waves the Lorentz force self-action on the matter field is zero when $E=B$, i.e. when real parts of the squared electromagnetic fields invariants ${\lambda{}}^2$ and
${\bar{\lambda{}}}^2$ are zero.
Particularly, this applies to electromagnetic waves in "vacuum" ($\bf{E}\perp{}\bf{B}$, $E=B$), where we have $\lambda{}=\bar{\lambda{}}=0$, and matter field equations coincide with Maxwell equations for "source-free" electromagnetic plane waves. In this case the momentum density $P^{\mu{}}$ of the matter field is non-zero, while the charge density $J^{\mu{}}$ is zero. In this sense the photons are not actually "source-free" electromagnetic waves.
For more details read the main article, where this model is fully described and applied to 3 types of stable elementary particles:

photons
charged fermions, and
neutrinos

QM with complex eigenvalues


Nokia Lumia 925 network selection problem

Question: I updated my Lumia 925 to Denim, but a strange thing happened after the upgrade; I can't select 3G. The drop-down list freezes on 2G. How can I fix this?
Comment: Have you tried getting your network to push the APN settings to your phone?
Answer: As Apoorv mentioned you could try and make a reset. But before that you should:

Check if there is another update available. Including installing and running Upgrade Advisor from Store.
Remember to have backup activated before you do.
Take note that there are two kinds of resets that you could do: Hard and Soft (always try soft first). You can read how to do both kinds of resets here.
Answer: Why dont you try to reset your phone ! Try to reset it and then see. Sometimes the firmware do creates problem so resetting is the only fix


Missing Samba upstart job

Question: Somehow my K20samba4 was deleted.
How can I restore it (ie does anyone know where I can get the script)?
Answer: I don't think we ship a K20samba4 script anymore - samba 4 now uses upstart.
There is a /etc/rc4.d/S20samba4 file, which should be a symlink to ../init.d/samba4
update-rc.d should also be able to create it for you.

Missing Samba upstart job


Is it possible to force someone you follow to unfollow you on Twitter?

Question: Is it possible to force someone to unfollow you on Twitter? Use-case: I want to follow my ex-girlfriend, but I don't want her to follow me. If I block her, I guess I can't see her tweets. Also, I would rather not use any 3rd party tools like follow her on RSS and block her.
Answer: You could block her, which I think will remove her from your followers list.  Then you could unblock her.  Of course, at that point she can follow you again, but it seems only fair that if you can follow her tweets, she can follow you.
Comment: Block indeed works this way; it breaks both follows and prevents the other person from following you. Unblocking doesn't restore the follows, it simply allows the other person to follow you again. So block-unblock should do the trick.
Comment: let's hope she doesn't read this...
Comment: Thanks for the tip, but not sure if block works that way. Also, nothing is fair with this girl.
Comment: But she can see if you start following her again if she has the notifications set to alert her when someboby new starts following her.
Comment: If you want that person not to follow you again, just keep that person blocked.
Answer: No you cannot. Twitter is an application for sharing. She has a right to follow anyone she wants if the person allows them to. Blocking is the only way to technically prevent her from following you.

Here is a suggestion of what you can do instead.
You can change your settings to protect your tweets

Only let people whom I approve follow
  my tweets. If this is checked, your
  future tweets will not be available
  publicly. Tweets posted previously may
  still be publicly visible in some
  places.

Then add her to a list.
Allowing for a quick link to her profile.
Comment: Thanks, but I would rather just force her not to follow me. Isn't it possible to force someone to unfollow you on twitter?
Comment: No it is not possible to do that. That would be [counter-twitter](http://twitter.com/about) like. You are just going to have to deal with it.
Comment: ok thanks. I'm pretty sure this was possible before.
Comment: No it was not. It was possible by exploit to force follow someone using `accept username` not the reverse.
Answer: You can be sneaky about this - create a new twitter account to follow her, then block her on your main account.
It's not possible to control who follows you outside of the block tool.
Answer: You can block someone and still view their tweets so long as they don't block you back but you have to manually go to their profile and click view tweets each time. That's really the only sure-fire way to keep her from following you while still seeing her tweets (which I assume is the reason you would want to continue following her) or you could create a second account and just follow her on that
Answer: Yes, you can force unfollow Twitter followers. I use Circleboom Twitter and it really works for me. It can detect inactive, overactive, fake, spam, and bot Twitter followers. Then, you can remove these followers in bulk on Circleboom.


What word is the opposite of Luddite?

Question: Single word, please, preferably one derived from someone's name.
Comment: Are you looking for the accepted antonym, or feeling out apt possibilities?
Comment: Meaning someone who positively embraces, encourages and implements industrialisation?
Comment: Sorry Barrie - I agree with the others that it's Too Localised. Or even Not Constructive, since you can only expect to get a load of neologisms. To which I suppose I could add [Gatesian](http://en.wiktionary.org/wiki/Gatesian), and [Jobsian](http://en.wiktionary.org/wiki/Jobsian)
Answer: Since Luddite describes the followers of one Ned Ludd who lead a movement against industrialization and in particular mechanized looms in 1811 or so, I will propose Cartwrightian since Edward (Edmund) Cartwright patented the power loom in 1785 and so his designs or variations thereon would probably have been the ones being destroyed by the Luddites.
But that may be too obscure for your use so technophile may be more appropriate.
Comment: Yes, I think Cartwrightian will do for my purpose and you seem tohave got there before crisis.sheep (actually an inquiry from someone else).
Answer: The problem is, the Luddites were named after an individual who smashed machinery, whereas their opponents were the state or at least the businessmen of the time. So the original antonym would have been Industrialist. You could identify a specific individual who opposed them, but they never had a specific anti-luddite movement, so it would not be entire accurate.
If you are looking for modern antonyms, technophile is probably the best, but again, there is no single individual who is leading the drive for technological progress, so it is not reasonable to associate a name with the pro-technology movement.
Comment: _"there is no single individual who is leading the drive for technological progress"_ For what it's worth, as I read that, Steve Jobs popped into my head. (Not that that helps solve the Luddite antonym problem, but I thought it was worth a mention nonetheless.)
Comment: @J.R. - there are a few significant names, Jobs being one, that relate to Arkwright or Cartwright in other posts. But there are others - Berners-Lee, Gates, Bezos, for example - that specific technophobic Ludditism might be focused on. It is the same problem as the original Luddites.
Answer: "Technocrat" appears to be the best fit here.
Otherwise I propose "Arkwrightian", after Richard Arkwright, inventor of the water frame that contributed to the onset of the Industrial Revolution (specifically because Ludd is reputed to have destroyed knitting frames, as Wiki has it).
Comment: Sorry, if I read the phrase "a modern-day Arkwright", I would think of the old comedy "Open All Hours".
Comment: Arkwright occurred to me, too; although I don't think one can make an adjective out of it. I think one would need to speak of "a modern-day Arkwright" -- and hope that readers' historical knowledge is up to interpreting the reference!
Comment: @DavidWallace Exactly :)
Comment: That was before my time, alas!
Answer:
Immanuel Kant (1724–1804) argued that progress is neither automatic nor continuous and does not measure knowledge or wealth, but is a painful and largely inadvertent passage from barbarism through civilization toward enlightened culture and the abolition of war. Kant called for education, with the education of humankind seen as a slow process whereby world history propels mankind toward peace through war, international commerce, and enlightened self-interest

I would say that Kantian is the word you are looking for. There are a lot more people following the Idea of Progress theory: John Locke, Voltaire, etc.
Comment: I'd expect Kantian to be a follower of [Kantianism](https://en.wikipedia.org/wiki/Kantianism).
Comment: Where are those citations from? Please tell us the name of where you got those from, and if applicable, also a link. If you are going copy out text verbatim, [our Help Center says that you must name](http://english.stackexchange.com/help/referencing) where you got the original from, and this post fails to do that. Please see the question on meta entitled ["What to do about missing source attributions: Copying, Linking, Attributions, and Plagiarism](http://meta.english.stackexchange.com/q/4973) for discussion on this.
Comment: The progress that is referred to in the quotation is not technological progress that the question is about. In any event, the term *Kantian* is already 'taken'.
Answer: Because Luddites are viewed as resisting dramatic technical progress, I would suggest, with no awareness of any such prior use, either:
Buckminsterite
or
Fullerite
Mr. Fuller was notable for his drive to expand both the creation of new technology and its innovative implementation.
Maybe we can start a movement.

What word is the opposite of Luddite?


Should I bother to reply to emails from recruiters when I'm already employed?

Question: Every now and then I get an email from a recruiter with a position they want to offer me. These emails are normally offering jobs outside my area of expertise and asking for skills I simply don't have. Not just that but I'm already employed in a job I'm quite happy in.
Answer: Typically, you can ignore recruiters who:

email you about a job that is far out of your area/experience
seem like they haven't done their homework on you
appear to be spamming anyone who matches keywords on LinkedIn or a job site

Ignoring these will not harm you, as they probably already forgot that they emailed you.
The good recruiters, however, actually do their homework and email you about jobs that are tailored quite closely to what you do. They are probably attempting to actually find the best fit, rather than rely on spam.
Replying to good recruiters can't hurt, since they may be a reliable asset down the line.
Comment: Keyword matching ... Once, a recruiter contacted me who was looking for people that have experience with "API". That tells everything.
Comment: @phresnel And then it turned out that they meant the American Petroleum Institute and it told even more... :-)
Comment: @phresnel Relevant: https://twitter.com/pjf/status/730215052750381056
Comment: I once was asked "how my XML programming was". After taking pity on the recruiter I proceeded to explain that XML was method of structuring data, not a programming language and I therefore couldn't attribute any skill level to my XML programming abilities. The response was "yes, but I need to know how good you can program XML"; the rest of the conversation was short lived
Comment: `probably already forgot that they emailed you` they didn't email you personally, its all automated.
Comment: @MasterScrat That was a human email though.
Comment: That's what I do, especially when they call me. I tell them I'm employed and that they can contact me again when my contract ends.
Answer: As a recent jobseeker completing a graduate degree and receiving many of these emails, I will say that most of these can be ignored for reasons already discussed. I am a PhD scientist and receive recruiting emails for door-to-door sales, bachelor's level lab technician work, and call-center positions. Frankly, these solicitations are not only irrelevant, but insulting. No reply warranted.
However, for recruiters with legitimate interest in you as candidate, it is worth replying in order to open up a potential networking opportunity. If they understand your experience, background, or were referred to you by a mutual colleague, it is worthwhile to reply and start a dialogue whether or not you are happily employed. You might know someone else who might fit the recruiter's needs and build a connection that way so that you might leverage that connection in the future.
Comment: To take this one step further, many recruiters will offer a bonus for referrals. Having a few good recruiters handy might make it easier to find work for your buddies + get you a cash payout on top of the networking opportunity.
Answer: I get emails from recruiters all the time.  Many times they have nothing to do with my skill set or my physical location1.  I ignore them totally.
My experience is that these emails are being generated by low level recruiters using shotgun approach2.  They send out bulk emails and hope that someone would actually fit their requirements (and also not complain when being low-balled on renumeration)
I also consider that recruiters who take this approach don't really care about you as a client but instead only care about counting up resumes that they can send to the company and thus maximize their own income.  As such I suggest that you form relationships with recruiters in your own physical area with who you can easy meet and can trust.

1. I develop software for industrial processes.  I once had a string of emails suggesting that I was the perfect Insurance salesman.  I never could figure that one out.
I also get lots of emails suggesting that I am a perfect web developer - nothing of which is indicated on my resume.
2. I once had two people from the same recruiter send me the same details about a single job (in this case the job was interesting to me and in my field).  These two actually sat across from one another in the same office.
Comment: "I develop software for industrial processes. I once had a string of emails suggesting that I was the perfect Insurance salesman. I never could figure that one out. " I work in IT too. I once got a spam about being a lifeguard ... several hundred miles from where I live.
Comment: Yeah, I'm in California, and I keep getting emails about gigs in New Jersey.  Don't they have Google maps back there?
Comment: ... renumeration?
Comment: Peter M: You say "I once had two people from the same recruiter send me the same details about a single job".  Once? It happens to me all the time.
Answer: You should reply if you want to receive endless recruiter emails since by replying you identify yourself as a "live prospect" and your contact information will end up in countless databases where it will be shared and even sold.
When you are looking for a job, if you decide to use a recruiter you want to be choose one based on recommendations from trusted colleagues in your field and on your own search criteria. Letting the recruiter choose you is not in your best interests.
Cheers!
Comment: Kind of a big [citation needed] even though I like to think this is all correct.
Comment: Agreed. I see a lot of folks recommending only to reply to ones that "did their homework on you" and I disagree with that since if you reply you'll get bothered endlessly because it's all automated and a chance they just accidentally sent something that you believe they gave thought to. That's why it's called "spam" because 10% of the time it produces a relevant information that a user believes is tailored to them so they act upon it.
Answer: I have never found a job that I got hired through a 3rd party recruiter. Most times the job was severely outside the scope of my skillsets or the recruiter would lie and when I went to the interview it was something completely different.
I've had very good luck just applying directly to positions. Just ask if they are a third party recruiter. If they answer NO, then ask them if they work directly for the HR firm for the company you would be working in if hired.
Even getting straight answers from these sleaze balls is hard.
Answer: You well receive many, many email automatically. If you don't reply, nobody cares. Nobody loses anything or wastes any time. If you reply that you don't want that position offered, you waste your time, and you waste the time of the recruiter.
Comment: If your reply, you **don't** waste the time of **some** recruiters, since he now knows that he has a valid e-mail to which future "correspondence" may be directed... At least, that's my thinking.
Comment: You should at least mark them as spam.
Answer: I recommend:

sending a quick canned reply explaining what areas you are interested in
keeping a list of all recruiters who contact you for what type of job on a text file. Never know when you might be out looking for a job again, and the list might come in handy.
Answer: There are a lot of spam recruiters out there who just send out mass emails to everyone they've seen a resume for without bothering to check if they might actually be qualified.  Some of the more obvious flags are they don't bother giving you the name of the company they are recruiting for (usually means they are a recruiting firm that knows they aren't a critical part of the recruiting process and is afraid you will go behind their back), they just list a few generic skills without describing the actual job (might mean there is no actual job, they are just trying to rope you into a conversation), your skills are completely inappropriate for the job (means they haven't done their homework), or sometimes even leaves parts of the template in the email (literally reading "Dear ").  If they are just emails, they are easy to deal with by flagging them as spam.  Such recruiters aren't useful as contacts if you really are looking for a job.
It gets really annoying when they also call you on the phone, especially during working hours...
Comment: In the modern world I have one simple rule. If you are not on my contacts I don't pickup the phone, meaning I don't pickup the phone for phone numbers that I don't recognize. I worry sometimes that it could be the doctors, hospital, police, or family emergency and fall into the trap of picking up the phone and I'm always wrong and it turns out to be an unwanted call. If it's important they will leave a voicemail.


Kammas are imponderable?

Question: Buddha says if one plants mango, he will reap only mango not something else. Short-lived and get killed are the result of killing living beings, poverty is the result of stealing, not trusted by others is the result of lying and so forth. Then why Buddha says the result of kamma is one of the four imponderables. Does it mean one who loves to kill living beings will experience poverty or not trusted by others which is the result of stealing and lying? How do we relate that with "sow apple seed, reap apple fruit?"
Answer: The kammas are imponderable because people tends to expect action 'A' resulting in the equal fruit 'A'. However as we know a single action 'A' is only one cause, there are many other conditions (other causes) which result in a fruit. Just as a cup of coffee implies the existent of coffee beans, but coffee beans by themselves do not necessarily result in a cup of coffee. There are so many other factors such as farmers, the sun, customers wanting to drink coffee for the outcome 'a cup of coffee' to result.
In my temple we like to say that the merit of a good deed is 'immeasurable'. Why is this? Because the outcome cannot be entirely predicted, a single good action, can lead to opportunities for further good actions to occur. For example, you help someone with a small task, and then that person gives you an opportunity to do more good, and eventually eventhough the original good action was only the beginning seed the final result is immense.
The only thing we know for sure is that good actions will result in good outcomes (if something consistently results in bad outcomes it is a bad action and you should stop doing it). So instead of trying to predict the final outcome, you should focus on the simple strategy of doing more good as opportunity allows.
Addendum regarding the comment
My understanding is that because the Four Imponderables can not be deduced logically therefore it is imponderable. For example a common stated effect of karma is said that killing leads to a shortened life span. There is no clear logical explanation why this is the case, you could argue it's through some kind of revenge, or perhaps susceptibility to illness caused by psychological damage, but it's only speculation.
Likewise, releasing animals doomed for butchering are said to be good karma for longevity, but once again there is no logical explanation why it would work.
Since speculations about the exact working of karma does nothing to lead to peace and end of suffering, the Buddha ask you to refrain from pondering about them and take it at face value.
Comment: I'm not trying to predicting anything but just want to get a little bit of picture to what extent we should understand the unconjecturable. You're saying there are many factors needed to produce a result but they (farmers, sun, etc.) are only supporting factors, right? The final result for killing doesn't change i.e. short life, that is known to us, so why it's imponderable?  Are you saying short life can be caused by killing and also can be caused by other causes. But poverty can never be the result of killing. Do I understand you correctly?
Answer: Not all Pali suttas are the words of the Buddha, which is why there are contradictions. In the Maha-parinibbana Sutta, this issue is addressed by the Four Great References. The scriptures also teach the teachings of the Buddha are to be verified by each wise person for themselves.
Personally, I am not willing to accept the Buddha spoke the Cula-kammavibhanga Sutta because what is written in it is not true. For example, rich people are often rich due to greed & unethical business practises where as poor people are often poor due to political or social oppression or exploitation. It is not because of past kamma that children are abused, raped or murdered. It is not because of past kamma that millions of people were randomly murdered by communists in the Soviet Union or China.
As for the imponderables in the Acintita Sutta, I guess this might mean the 'exact' workings of kamma in minutiae detail. Kamma is a general rather than exact principle. Kamma is not ultimate truth, which is why in the Maha Kammavibhanga Sutta it is said good karma can also lead to a bad result & bad karma can also lead to a good result.
Comment: @Dhammadhatu So do you agree a person who loves to kill living beings will experience poverty, not trusted by others, etc., as the result of killing living beings?  From what I understand, good kamma will **always** lead to good result. Bad kamma will **always** lead to bad result. But when will it give the effect, we do not know. Regarding rich people, everyone has greed but not all greedy people are rich. Being born in a wealthy family, for example, is one of the "right condition" for the good kamma to take effect, the same thing as being poor and socially oppressed.
Comment: Note the Cula kammavibhanga never says that all rich people are so due to being givers -- which is very suspect indeed. It also never says that a rich person is so due to giving things in its current life. What it says is that, a person who is a giver *now*, when *dying*, if he or she reappears in human realm, he or she will find him/herself born rich -- a plausible reading would be being born in a rich family. The same applies for other qualities.
Comment: Kamma is not ultimate truth, as I posted from the suttas. MN 117 states the moral right view of kamma is tainted (defiled)by the burden (upadi) of self-view. I can do good karma by loving & raising a perfect successful family. But if my family die in an accident, I will suffer. This shows good karma does not always lead to a good result. I can do evil & learn this is harmful. From knowing clearly the harm of that evil action, I can shun evil & thus become enlightened. There is no absolute lawfulness of karma, as the scriptures state.
Answer: Lord Buddha himself said that karma is like a ripple effect.
Let's assume you killed the person A by firing a bullet through the forehead. So in direct terms, you should also be fired a bullet through your head. But it may and may not happen. Let me tell you why.
Now when you killed that person A. There is a chance that you caused these effects on him :

Caused him physical pain before death
Caused his family immense pain
Caused his friends pain

Now you have caused person A's family a lot of heart breaks. Imagine, because of that person A's death, his spouse and children have no way of earning money and gets mentally weakened. The children do not get proper education nor the guidance. So the children end up drug addicts. Now the children are addicted to drugs and they grow up and commit murders to find money for drugs. Person A's parents die in terrible sadness and shock because of their sons murder.
This is why the Karma is like the ripple effect, it follows a lot of things. Hence you ignored the consequences of your actions to the people around and you just shot person A in cold blood. Now the sins you pay will be determined by these things.

If you intended to kill person A
If you knew the person A had living parents
If you knew the person A had a spouse and had children

If all of the above is positive and committed the murder with the
intention of hurting the entire family, then you have committed a
massive sin and you will pay for it for a very longtime in hell.
If you only wanted to kill him out of anger but did not know he had
living parents or family. The effect will most likely be
less than the previous one but still you will pay for a very
longtime in hell.
If you didn't mean to kill him and it was totally some sort of an accident then probably there won't be any bad karma on you.

For the first two situations, the way you will pay for these may and may not happen like an mirror image of what you did. Mostly likely it won't. Because that is not the only Karma you have with you. You have previous good karma and bad karma. So the good and the bad karma will determine how you pay for each one.
Answer: I would like to ask the rebuttal question to this? Are we afraid of the consequence of our actions?
Yes karma will deal with us and the future. Possibly. But yes we will never know what exactly we have done and to what consequence. Because we have not reached that level of perception.
If we do much Sadhana, we may reach that level, but then again I am nobody to ask you to do that.
Having said that, let's try to be aware of every moment, mindful, and let's face what the future has for us.

Kammas are imponderable?


Why does the Talmud record the Rabbis' thoughts and theories about the science of natural phenomenon?

Question: There are various areas in the Gemara where there is discussion about natural events, often appearing like a tangential discussion. Some examples are in Bechoros 7-8 or Pesachim 94. I understand that many halachic or historical topics are arranged in this tangential manner, but what is the reason for including hypotheses about science? If someone thought thunder is caused by lightning breaking apart balls of hail and another thought shooting stars were rips in the firmament that let the light from behind it through or another argued that the Greeks were wrong about the duration of a viper's gestation or another suggested the sun travels West to East behind the firmament at night, why include their suggestions on such subjects? (Note that I'm not asking about areas where there is actual halachic implications to the topics as with discussing if Tamei fish lay eggs or what time of day a chicken can lay an egg.) It appears that including such discussion risks opening it up for criticism about the Rabbis' knowledge, so there should be some positive purpose for including it.
Comment: Maybe no one expected future Jews to read it as prophecy? They included interesting information from wise people.
Comment: It's also worth noting that people turned these into pseudo prophecies as a response to reformers who were leaving the words of Chazzal for a modern irreligious  world view.
Comment: related http://judaism.stackexchange.com/q/30509/4794
Answer: According to the Maharal, amongst others, literal readings of non-Halachik Talmudic or Midrashic statements are incorrect and indeed Chazal are not discussing natural phenomena at all. Rather, they are discussing fundamental metaphysical realities and theology at a level that requires a deeper understanding of their language and philosophy (See, e.g., Maharal Be'er HaGolah Be'er 6 where he discusses e.g. the passage in Pesachim you cite; See also e.g. Maimonides' Introduction to Commentary on the Mishna and Vilna Gaon's commentary on Mishlei 1:3).
Comment: It's worth noting that the Maharal [along with the Geonim, Rambam (probably), and Vilna Gaon] was one of those opposed to the calculation  of *Shekiat HaChama*, based on a literal reading of Pesachim 94a, as later than actual sunset
Comment: Does he provide support for that position? There are certain things that I find it very hard to read non-literally. Further, would that not apply to historical details as well? My understanding was that mainstream Orthodoxy believes the things the Gemara says about events like Purim or the destruction of the second temple.
Answer: There is a letter from Rabbi Hirsch found here http://www.mesora.org/hirsch.html now printed in the ninth section of his collected writings.
This letter deals with statements of Chazzal which deal with their view of sciences and seem bizarre to the modern mind.
Here's two excerpts:
Sages of Torah, not Masters of Science
In my opinion, the first principle that every student of Chazal's statements must keep before his eyes is the following: Chazal were the sages of G-d's law - the receivers, transmitters, and teachers of His toros, Hismitzvos, and His interpersonal laws. They did not especially master the natural sciences, geometry, astronomy, or medicine - except insofar as they needed them for knowing, observing, and fulfilling the Torah. We do not find that this knowledge was transmitted to them from Sinai.
Later he says:
The Talmud in Bova Kama declares "A human spine, after seven years, turns into a snake; this applies only if he did not kneel atModim. " Anyone who reads this finds it laughable, but Pliny says the same statement almost word for word, "After a number of years the human spine turns into a snake." Chazal, however, used this to teach a mussor lesson. To any mind it is clear that every similarly surprising statement of Chazal, if we look into it, was accepted as true by the scholars of the time.
So there are two important lessons Rav Hirsch is pointing out from these Chazzal.
 One, don't ignore the reality as explained by contemporary scientists, deal with it.
 Two, the lesson learnt is the point, not the lesson plan or the message used.
Comment: The gemara in Bava Kama is 16a. And just for the record, this letter's authenticity has been questioned by some right wing Rabbi's who just so happen to disagree with the message therein.
Comment: What you appear to say is that even when suggesting scientific theories, what they meant was to get across lessons, regardless of what they knew of the natural world. But this does not apply to many if not most of their scientific theories. Or does it? If not, what is the point of those?
Comment: @A L that was one point. I'm not sure exactly how Rabbi Hirsch would explain many of the ideas brought in Shas. Take your comets ripping the firmament one. I don't get the mussar in that one, but Rag Hirsch maintained this point. Personally, I have a slight twist on this. Imagine if all the scientific points they raised were proven right. Wouldn't we then have so much more respect for them and their Torah knowledge? What about all the rabbis nowadays who are up to date on the sciences? Don't so many people respect them more due to that? Perhaps Chazzal were trying to impress upon the people
Comment: That they were up to date and knowledgeable so as to get the people to listen to them and keep the Torah properly.
Comment: @A L I suggest you read that letter in its entirety. But there are always going to be the weird oddities like when Chazzal talk about Romulus and Remus and stuff like that.


If someone was charged with 15 counts but was only indicted on 2, can the prosecutor introduce evidence at sentencing of charges that we not indicted?

Question: I just have a question about information being presented that a defendant was not indicted on. If someone was charged with 15 counts of a crime but was only indicted on 2 counts, can the prosecutor introduce evidence at sentencing of charges that the person was not indicted on? And if so isn't that denying the defendant his or her due process because that person was not indicted on those charges?
Answer:
If someone was charged with 15 counts of a crime but was only indicted
  on 2 counts, can the prosecutor introduce evidence at sentencing of
  charges that the person was not indicted on?

In federal court, yes. This has been the case since Williams v. New York, 337 U.S. 241 (1949) which held that evidence such as counts and conduct upon which the defendant was not charged or indicted may still be considered in a sentencing hearing. Since U.S. v. Watts, 519 U.S. 148 (1997), the prosecutor can even introduce evidence at sentencing of charges upon which the person was acquitted by a jury. See generally here. This remains good law although it is controversial both in academic writing and in political discussions and among many sitting federal judges.

Neither the Federal Rules of Evidence nor constitutional provisions
  related to evidentiary matters (e.g., the Confrontation Clause of the
  Sixth Amendment) apply at sentencing. Therefore, the court may
  consider hearsay and other types of information that would not be
  admissible during a trial. However, the [U.S. Sentencing] Commission
  has stated that information considered by a court at sentencing must
  have "sufficient indicia of reliability to support its probable
  accuracy." Under Federal Rule of Criminal Procedure 32, the court
  "must — for any disputed portion of the presentence report or other
  controverted matter — rule on the dispute or determine that a ruling
  is unnecessary either because the matter will not affect sentencing,
  or because the court will not consider the matter in sentencing." In
  resolving factual disputes, the court ordinarily applies the
  preponderance of the evidence standard.

From here. The quote above cites the following authority:

FED. R. EVID. 1101(d)(3).
Williams v. New York, 337 U.S. 241 (1949).
18 U.S.C. § 3661; see also Pepper v. United States, 562 U.S. 476, 480
  (2011) ("This Court has long recognized that sentencing judges
  'exercise a wide discretion' in the types of evidence they may
  consider when imposing sentence and that '[h]ighly relevant — if not
  essential — to [the] selection of an appropriate sentence is the
  possession of the fullest information possible concerning the
  defendant's life and characteristics.' Williams v. New York, 337 U.S.
  241, 246– 247 (1949). Congress codified this principle at 18 U.S.C. §
  3661, which provides that '[n]o limitation shall be placed on the
  information' a sentencing court may consider 'concerning the
  [defendant's] background, character, and conduct,' and at § 3553(a),
  which sets forth certain factors that sentencing courts must consider,
  including 'the history and characteristics of the defendant,' §
  3553(a)(1).").
USSG §6A1.3(a).
FED. R. CRIM. P. 32(i)(3)(B).
USSG §6A1.3, comment.; see also McMillan v. Pennsylvania, 477 U.S. 79,
  91-92 (1986) (holding that due process does not require sentencing
  factors to be proved by more than a preponderance of the evidence).

Federal Rule of Evidence 1101 states (emphasis added):

Rule 1101. Applicability of the Rules
(a) To Courts and Judges. These rules apply to proceedings before:
· United States district courts;
· United States bankruptcy and magistrate judges;
· United States courts of appeals;
· the United States Court of Federal Claims; and
· the district courts of Guam, the Virgin Islands, and the Northern
  Mariana Islands.
(b) To Cases and Proceedings. These rules apply in:
· civil cases and proceedings, including bankruptcy, admiralty, and
  maritime cases;
· criminal cases and proceedings; and
· contempt proceedings, except those in which the court may act
  summarily.
(c) Rules on Privilege. The rules on privilege apply to all stages of
  a case or proceeding.
(d) Exceptions. These rules — except for those on privilege — do not
  apply to the following:
(1) the court's determination, under Rule 104(a), on a preliminary
  question of fact governing admissibility;
(2) grand-jury proceedings; and
(3) miscellaneous proceedings such as:
· extradition or rendition;
· issuing an arrest warrant, criminal summons, or search warrant;
· a preliminary examination in a criminal case;
· sentencing;
· granting or revoking probation or supervised release; and
· considering whether to release on bail or otherwise.
(e) Other Statutes and Rules. A federal statute or a rule prescribed
  by the Supreme Court may provide for admitting or excluding evidence
  independently from these rules.

Given the immense personal importance of a sentencing hearing which can exercise discretion over a range of many decades of possible prison time, and which is frequently the only meaningful opportunity for a person who is easily proven guilty to distinguish himself or herself from other defendants, it is really very stunning and counterintuitive that this is the case.

And if so isn't that denying the defendant his or her due process
  because that person was not indicted on those charges?

Under current U.S. Constitutional law, the maximum sentence that can be imposed must be based upon convictions by a jury or judge based upon charges upon which indictments were brought, which were proved beyond a reasonable doubt based upon admissible evidence. The leading case in support of this proposition is Apprendi v. New Jersey, 530 U.S. 466 (2000).
But, at sentencing, the judge's exercise of discretion up to the maximum sentence established at trial may be made based upon a preponderance of the evidence and that evidence is not subject to the same procedural safeguards.
Comment: The question asked about counts that were not found to be indictable, not those where there was an acquittal.  I'm not disputing that case comes into play after an acquittal, but when counts were dropped, the court rule I mentioned controls it.
Comment: Also, why would the judge ask if the defense would like to object if it is always allowed? He or she wouldn't, so this does not apply.
Comment: @Putvi The federal rules of evidence specifically state that they do not apply to sentencing hearings. The only objections allowed in sentencing hearings are for privileged information and that the evidence lacks "sufficient indicia of reliability to support its probable accuracy." A judge may respond by either ruling on the objection on the merits or by asserting that the judge isn't considering the disputed evidence in any case rendering the objection moot, as noted in my answer. (A defense attorney may also preserve an objection seeking to change the law for appellate review by objecting).
Comment: @Putvi One can also imagine other objections that don't arise from the rules of evidence. Perhaps the prosecutor is jumping the gun and starting to present evidence before the incarcerated defendant is the courtroom (which isn't allowed). Perhaps the prosecutor is reading from a sentencing report for a co-defendant instead of the person whose sentence is being considered. Perhaps the manner of an evidentiary presentation is so abusive and insulting of either the defendant or his counsel that the defense wants to shut it down as contemptuous. The FRE don't apply merely because one can object.
Answer: It really depends on how the information is used and what the information is. It is impossible to say without knowing the exact circumstances, but I will give you an overview.
Generally, if you want to use it to only cast the person in a certain light, it not permitted.
Federal Rule 404 says:

(2) Permitted Uses; Notice in a Criminal Case. This evidence may be admissible for another purpose, such as proving motive, opportunity, intent, preparation, plan, knowledge, identity, absence of mistake, or lack of accident. On request by a defendant in a criminal case, the prosecutor must:
(A) provide reasonable notice of the general nature of any such evidence that the prosecutor intends to offer at trial; and
(B) do so before trial — or during trial if the court, for good cause, excuses lack of pretrial notice. https://www.law.cornell.edu/rules/fre/rule_404

In some cases, the other acts would show that something was planned or intentional and is admitted. So, if the information is needed to show something, it is allowed, but it is not allowed solely to cast a person in a certain way.
Comment: The federal rules of evidence do not apply in sentencing hearings in federal court. See FED. R. EVID. 1101(d)(3).
Comment: Was the sentence high?
Comment: Was it a serious crime though or the person did whatever it was a lot?
Comment: Sometimes one serious crime is all it takes tbh. Not that I wish bad on the person.


How important is memorizing chess openings in casual play?

Question: I've played chess since I was kid and after much practice have become a decent player.  I still though, have little strategy to my opening moves besides starting with a King's Pawn Game.  I know professional players and grandmasters have a large repertoire of openings they know.  They also know a large number of responses to those openings.  How important is knowing different openings when playing chess casually?
Answer: Its far better to thoroughly understand a small number of openings than it is to mindlessly memorize a large number of openings.  Frequently, if you know why a set of moves is considered optimal, then you're in a better position to adapt once the board gets "off the opening".  However, if you have a wide set of openings memorized, you may find that while you are moving down your 'optimal opening', ready to transition to a strong midgame, your opponent may have other plans, and make a move that takes him off your chosen opening.
Now frequently, there's a good reason why that move is not covered by your opening chart;  that move may be terrible.  But if you don't understand why that move is terrible, then you may not be able to take advantage of it, and if the other player improvises better/faster than you, you may find yourself at a disadvantage despite your 'superior' play to this point in time.
Don't let that happen to you.
Comment: I don't think our viewpoints are that far off;  you seem to be saying "Understand tactics over opening theory", whereas I am saying "Don't memorize openings, understand them".  We both seem to agree that rote memorization is a bad idea, at least.
Comment: I'm going to have to respectfully disagree with this answer - if you are a casual player, even studying opening *theory* is a waste of time.  **That time is much better spent studying tactics.**  Even at only half an hour a day doing puzzles, it is fairly easy to jump from 1000 to 1400 in only a few weeks, a **huge** improvement.  If you spend that time studying openings instead, you will likely find little to no improvement in your play.
Answer: 1000x more important (and more fun!) than even studying theory is.. improving your tactics!  The best place I know for this is chesstempo.com.
You can easily get up to 1700~1900 USCF never learning any theory at all (casual players are usually 800~1400), and I've even met a few players over 2000 (master level) who have never studied any theory!
However, if you insist on studying theory, learn the barebone basics, then study endgames - knowing an opening might give you a slight theoretical edge going into the middlegame, but knowing an endgame could mean the difference between winning and drawing/losing. Remember, the winner is the person who makes the second-to-last mistake!
Comment: This answer may have some excessive formatting.
Answer: For casual play, I don't think memorizing chess openings is all that important. However, studying and understanding why some counter-moves are appropriate for this or that opening may prove quite useful.
Answer: Chess openings are very important if you are interested in winning games, as they are sets of moves that help develop your pieces quickly and gain some control of the centre of the board.  Having a grasp of an opening sequence that your opponent doesn't know should give you an edge in the early game that you may be able to turn into a clear lead during the middle game.
On the other hand, if you are playing against an opponent who doesn't know the opening then you may quickly find yourself in an unexpected position - so try to concentrate on learning a shallow amount of an opening.
Finally the other thing to say is that some openings are a must know.  See for example scholar's mate!
Comment: I'm quite familiar with a variation of scholar's mate that we used to used to try on each other in middle and high school to see if you were paying attention.
Answer: Opening memorization is not important at all, especially to casual play, and shouldn't be the focus of study time for serious players, either, even up to around expert/master level.   For casual play, you should instead follow basic opening principles: put a pawn or two in the center, move all your pieces once before moving any piece twice (unless there's a tactic), castle.   For serious players, it's worth playing a few openings and learning them well, slowly building your mental tree of opening variations game by game, by playing a game and looking up the official lines afterwards, to see where you diverged.
I once spoke with a well known local expert player, who confided to me that he wished he could "get all those years back" that he spent studying openings.   I asked "what would you study instead?"  He instantly responded: "tactics and endgames."   He usually played unorthodox gambit openings and almost always built a great attack with the initiative.
Answer: You don't need to memorise every opening move and variation to be able to play it - not for casual play at least. It's better to just play and get a feel of the different characters that varied first moves and common responses can give a game. You may find you enjoy losing games trying to crack a c4 first move more than winning with a King's Pawn game, and you'll learn to understand the opening better than you ever will by memorising the first X moves.
And let's face it, for most of us the game isn't won or lost in the first few moves; there are plenty more opportunities for middle and end-game blunders. A world of experimentation and catastrophic strategic mistakes awaits!
Answer: In casual play, it's not necessary to memorize openings. Because your "casual" (by definition) opponent won't know them.
It's much more important to know tactics, and to "read" the opponent's moves. If s/he departs from the "book," its usually to that player's detriment. Occasionally he'll stumble on an alternate variation, rarely find a new line. Figure out why your opponent's moves are (probably) wrong, and take advantage of that fact.


What is the biblical basis for salvation by faith alone (sola fide)?

Question: One of the key issues in the Reformation was the doctrine of Sola Fide, that is, that salvation is by faith alone, apart from works.
What, then, is the biblical basis for this belief?
Comment: I can't believe this question hasn't been asked here before.  I mean, I did a search, and it hasn't, and I'm absolutely amazed.
Comment: I agree that there appears to be more evidence of Faith and Works vs Faith alone in the bible. [James 2:17](https://www.biblegateway.com/passage/?search=James%202:17) and [John 14:12](https://www.biblegateway.com/passage/?search=John+14%3A12&version=NIV) come to mind.
Comment: From the counterpart of this question, one of the main points of the [top-voted and accepted answer](http://christianity.stackexchange.com/a/6915/10486) could equally have been applied as a comment here: 'salvation' should be replaced by 'justification' in this question, so that it more accurately reflects the actual doctrine of *sola fide*. Is it too late to do this now without upsetting the apple cart too much?
Answer: I'm going to have to limit my quotes from Scripture and leave some compelling verses out, because there are simply so many.  This answer would be even longer than my  normal long-winded answers if I included all supporting Scripture references.  Please forgive me if I miss something you consider relevant.

First, there is the entire book of Romans, which is a build-up to, statement of, and then supporting evidence of, the doctrine.  Some relevant verses:
Chapter 1 is a description of Gentiles (non-Jews) and shows the wickedness as part of the build-up.  By extension, this shows that religious affiliation (even to a so-called "one true Church" as many denominations claim) can save us.
Chapters 2 segues into showing that the Jew is no better, for even though they have the promise of God, they are still corrupt.
Chapter 3 continues to show that the Jew (and by extension everyone) is unable to attain salvation through the works of the Law and gets to verse 27:

Romans 3:27 and 28 (KJV)
27 Where is boasting then? It is excluded. By what law? of works? Nay:
  but by the law of faith.  28 Therefore we conclude that a man is
  justified by faith without the deeds of the law.

The remainder of Romans is support for this doctrinal statement.  Paul starts out by showing how the early heroes of the Jewish faith were not justified by their own goodness, but by Faith on God.
He then goes on to show how the Law is perfect, and cannot save us, but rather was given to us that we might know what sin is - so that we can realize that we are sinful.
In the words of Ray Comfort in his teaching on Hell's Best Kept Secret, God's Law is like a mirror that shows us to ourselves in our true nature.  If we see we have a dirty face in a mirror, we don't wash our faces with the mirror.  No, we wash our faces with water.
Likewise, we can't wash away our sins with the mirror of God's law, but instead are washed clean by the blood of Christ. This is the essence of the message to the Romans.
Other verses throughout the New Testament also state this doctrine plainly.

Epesians 2:8-9 (KJV) 8 For by grace are you saved through faith; and that not of yourselves: it is the gift of God: 9 Not of works,
  lest any man should boast.

As I said in the beginning, a comprehensive list of Scriptures would be prohibitively large, and it would be impossible to do a better job of answering this than has already been done by Paul in Romans, but here is a link to an article with more.
http://www.biblefacts.org/bible/salfaith.html
Comment: This answer would have been better if it actually did the extensive quoting, rather than generally referring to Romans and asserting that it supports the doctrine. Two quotes from the Bible is a rather thin showing of the *Biblical basis* of a doctrine.
Comment: Shouldn't the second sentence describing chapter 1 actually be in the section describing chapter 2? And isn't it missing a 'no' before 'religious'?
Answer: In addition to David's thorough answer, I would like to add (what I think) is the single most clear verse:

Galatians 2:16
We who are Jews by birth and not 'Gentile sinners' know that a man is not justified by observing the law, but by faith in Jesus Christ. So we, too, have put our faith in Christ Jesus that we may be justified by faith in Christ and not by observing the law, because by observing the law no one will be justified.

(I added verse 15 for clarity, verse 16 is in bold.)
The NIV Study Bible says:

Galatians stands as an eloquent and vigorous apologetic for the essential NT truth that man is justified by faith in Jesus Christ - by nothing less and nothing more - and that he is sanctified not by legalistic works but by the obedience that comes from faith in God's work for him, in him and through him by the grace and power of Christ and the Holy Spirit. It was the rediscovery of the basic message of Galatians that brought about the Reformation. Galatians is often referred to as "Luther's book", because Martin Luther relied so strongly on this letter in his writings and arguments against the prevailing theology of his day. A key verse is 2:16 (see note there).

And that note is:

A key verse in Galatians. Three times it tells us that no-one is justified by observing the law, and three times it emphasises the indispensable requirement of placing one's faith in Christ.
Answer: I think it would also be helpful to say that 'salvation by faith' is not something that only Paul talks about. He maintained that this had its bases in Old Testament scripture too when he quoted Genesis 15:6, 'And he [Abraham] believed in the LORD; and he counted it to him for righteousness' (AV/KJV). This is at a time way before the Law had been delivered to Moses, so that righteousness on the basis of the Law or good works could be discounted.
This of course becomes the basis of Paul's discussion in Romans 4:

What then shall we say was gained by Abraham, our forefather according
  to the flesh? For if Abraham was justified by works, he has something
  to boast about, but not before God. For what does the Scripture say?
  "Abraham believed God, and it was counted to him as righteousness."
  (ESV)
Answer: In answer to the recent comment added, I will edit this answer to provide better explanation.

Luke 18:9-14.
  He also told this parable to some who trusted in themselves that they were righteous, and treated others with contempt: "Two men went up into the temple to pray, one a Pharisee and the other a tax collector. The Pharisee, standing by himself, prayed thus: 'God, I thank you that I am not like other men, extortioners, unjust, adulterers, or even like this tax collector. I fast twice a week; I give tithes of all that I get.' But the tax collector, standing far off, would not even lift up his eyes to heaven, but beat his breast, saying, 'God, be merciful to me, a sinner!' I tell you, this man went down to his house justified, rather than the other. For everyone who exalts himself will be humbled, but the one who humbles himself will be exalted."

Since salvation is salvation from God by God and related to the state of the relationship between God and mankind, then whether one is righteous (or in right standing with God) is what is the key issue. One is saved from the punishment of God, by God, via God's own provision of the means of being in right standing with him. That is the sin atoning work of Jesus, his propitiation, and imputation of his righteousness to all those that are his and who are to accept it (God's grace).
So the above passage is simply that, the means of being called righteous is that acknowledgement of one's sinfulness and the need for God's mercy. It is simply asking God to be saved from what one deserves. Humility, repentance and acceptance that it is God who saves.

'God, be merciful to me, a sinner!'

ESV. Luke 23:39-43.One of the criminals who were hanged railed at him, saying, "Are you not the Christ? Save yourself and us!" But the other rebuked him, saying, "Do you not fear God, since you are under the same sentence of condemnation? And we indeed justly, for we are receiving the due reward of our deeds; but this man has done nothing wrong." And he said, "Jesus, remember me when you come into your kingdom." And he said to him, "Truly, I say to you, today you will be with me in Paradise."

And again, the one criminal simply recognized that he was a sinner and recognized who Jesus was. By asking Jesus to remember him when Jesus enters his kingdom is a fairly profound acknowledgment that Jesus is not of this world, and that his kingdom is after death, and whatever Jewish milieu of kingdom understanding.
In both passages that I chose, someone is justified by their acknowledgement of who they are and who God is, and their relation to him.
There are countless resources providing scriptural basis for salvation by faith alone, hence why I didn't reproduce those. I chose the two passages simply because you have a clear presentation that someone standing with God is dynamically changed simply because of their acknowledgement of their relationship to God, not because they have done anything to gain favor.
Comment: Why do you think these verses are the Biblical basis? Please give an explanation or this is akin to a link only answer.
Comment: I agree with @curiousdannii that simply providing two Bible quotes, without any explanation of how these verses support the doctrine of salvation by faith alone, does not make for a solid answer to the question.
Answer: The book of Hebrews,  chapter 11, is all about faith. HEB 10:38 says "the just shall live by faith." To help connect, John 3:15 says, "That whosoever believer in him should not perish,  but have eternal life. " We believe in sunlight because we see it, experience it. But we also have faith that spring will come after winter, bringing warmth and sunlight. But biblically, as seen in Hebrews 11, faith is the substance of things hoped for,  evidence of things we don't see. HEB 11:6 says without it, we cannot please God. Vs 7 shows how Noah acted out of faith [Gr pistis], and became an heir to righteousness which is by faith. Also in Hebrews, whether accepted as doctrine or simile, we see in 11:31 that Rahab did not perish with the unbelievers having received the spies in peace, by faith.
Mark 11:22 says, "And Jesus answering saith unto them, Have faith in God." This portion of scripture goes on to talk about having what you believe,  by faith,  not doubting in your heart,  and that so doing you shall have it.  We could now go back Luke 23 and see the convicted thief who asked the Lord to remember him when He enters His kingdom...he believed, he had faith, and Jesus confirmed it with His words.
Acts 4:12 tells us there is no other name given men under heaven by which we must be saved.  Eph 2:8 says, "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:".  Not of ourselves, not by our power or by effort, works, or design; only by faith.
These are only a few references to help outline the concept. I must say here that if faith were on easily-described concept, humanity would not have the struggles we have endured through the ages. I do not believe that there IS any ONE verse that answers this question. But it did cause me to post when I saw the rightful argument that there was only a single reference used. After many many years of preaching and teaching, Paul said, "that I might know Him", understanding he had yet to plumb the depths of understanding to be had in faith in Christ and salvation.  In closing, I will give a scripture that keeps me studying to better understand:
Acts 17:11 "These were more noble than those in Thessalonica, in that they received the word with all readiness of mind, and searched the scriptures daily, whether those things were so."
In other words, don't take my word for it, find it. (all ref was from KJV)
Answer: Salvation is described in the Bible as through justification. Justification is recognition by God that you are his sheep, his family member.
Salvation is fulfillment, the rescue of a failed mission, to subdue the earth. It's a hard task, made possible only by help from God. However only the pure can see God, and Adam's culpability through being mentally competent (knowing good from evil) has separated him from God. God's plan is to remove culpability, restore the immunity, not by making humanity mentally incompetent like Adam was, but by making the New Humanity a mercy seat, through Messiah. Being in the New Man in Christ would cover sin, make a person pure, able to be united back with God. A weak analogy is diplomatic immunity. We are ambassadors. Now man can subdue creation. Manifest God's great works done through himself, so that the world can be blessed, by turning people to follow God, like Rahab.
When a believer exhibits signs that identify him as being on God's side, shows faith, loyalty, he is justified, baptised, marked, recognised as God's sheep, as family member. Testing confirms he is clean, and only requires washing of feet, detailing, topping up.
Why does works not justify?
Works is part of another agreement. When Israel was led into the desert, manifestations of God's great works scared her. She desired an arrangement that protected her from God's wrath. She would follow a predetermined list of rules, and God would cease and desist from wrath, raining down death and destruction, like He did, on Egypt.

Deuteronomy 5:24"You said, 'Behold, the LORD our God has shown us His glory and His greatness, and we have heard His voice from the midst of the fire; we have seen today that God speaks with man, yet he lives. 25'Now then why should we die? For this great fire will consume us; if we hear the voice of the LORD our God any longer, then we will die. 26'For who is there of all flesh who has heard the voice of the living God speaking from the midst of the fire, as we have, and lived? 27'Go near and hear all that the LORD our God says; then speak to us all that the LORD our God speaks to you, and we will hear and do it.'

God would have preferred following the promise He had made to Abraham, but agreed, seeing the list as a good way of helping His People in recognising Messiah, when He was sent.

Jeremiah 7:22"For I did not speak to your fathers, or command them in the day that I brought them out of the land of Egypt, concerning burnt offerings and sacrifices. 23"But this is what I commanded them, saying, 'Obey My voice, and I will be your God, and you will be My people; and you will walk in all the way which I command you, that it may be well with you.'
Romans 3:1Then what advantage has the Jew? Or what is the benefit of circumcision? 2Great in every respect. First of all, that they were entrusted with the oracles of God. 3What then? If some did not believe, their unbelief will not nullify the faithfulness of God, will it?

However, the new arrangement did not abolish the Promise, they coexisted. Believers could still be accepted as family, on showing faith, loyalty to God. Those who could try to be compliant could also be protected from God's wrath, if they flowed the rules perfectly. Ironically, no one could avert God's wrath or be justified by works, while it was possible to both avert wrath and be justified, through faith.

Galatians 3:17What I am saying is this: the Law, which came four hundred and thirty years later, does not invalidate a covenant previously ratified by God, so as to nullify the promise.

To answer the question, the Biblical basis for the doctrine of faith alone is that works could only protect from God's wrath, according to the agreement. There was no benefit in the Sinaitic Treaty of being recognised as family, salvation. Even protection from wrath, curses, was only conditional of perfect compliance with the 613 rules. Which no one could fulfill. Faith, on the other hand could both avert curses, as well as result in blessings, by being accepted as family.

Genesis 15:6Then he believed in the LORD; and He reckoned it to him as righteousness.

All Scripture from the NASB.
Answer: Jesus says in many places that blindly following the Law of Moses will not lead to salvation, and is his primary criticism of practicing Jews of the time.  Belief in Jesus' message and God is required for salvation.
Now, mainstream Christians take Jesus' statement "I did not come to abolish the Law but to fulfill it" as evidence that he has completed the Law and belief in him is enough going forward to attain salvation by faith alone.
However, some Christians (e.g. Messianic Jews http://en.wikipedia.org/wiki/Messianic_Jews), believe that Christians must believe in Jesus' message and God to attain salvation.  But they also follow the Law of Moses as they lead their lives, not because the Law is required for salvation, but because otherwise they will be committing sin given that Jesus did not abolish the Law.  Even these Christians believe in salvation by faith alone, but they do not excuse themselves from the Law believing that God will punish them for sins due to breaking the Law, before ultimately accepting them into Heaven.
There seems to be major misunderstanding of what Jesus says and preaches, and this shows in answers provided by others.  Most Christians believe that Jews are wrong to rely on practicing the Law of Moses as a pathway to salvation.  This is not the case.  As far as I know, Judaism also believes in salvation by faith alone.  The Law is to be followed to avoid displeasing God and not sin.  Following the Law to the letter, without faith in the One God, is not enough.
Jews do not believe that abiding by the Law of Moses is enough for attaining salvation - belief in the One God is key to attaining salvation.  Jesus saw that many Jews of his time were obsessed with details of the Law and its interpretation, and not on Moses' call to focus on the belief in the One God.  Jesus came to correct them in their practice and reinforce belief in God.
I am not even close to being knowledgeable enough to answer the question in a scholarly manner, but the above difference in belief is what I have uncovered by practicing in discussions on this portal.  Others may be better able to elaborate further.
Comment: I know of at least one answer you're talking about (I provided it) but it would be helpful to others if you linked to them.
Comment: The question was about the Biblical basis for salvation by faith alone. This answer makes some general statements, but doesn't back up the critical ones with suitable quotes, such as where the Bible actually says these things, or statements by key teachers in Messianic Judaism saying that they believe in salvation by faith alone. Links are not sufficient to make the argument.


Evolution Install

Question: I'm trying to install Evolution and the Ubuntu Software manager tries to install it and just stops. Evolution gets pinned to the launch bar but won't start.
I then tried from the command line using sudo apt-get install evolution evolution-mapi but that fails with the following output:
The following packages will be upgraded:
  gnome-icon-theme gnome-icon-theme-symbolic libasn1-8-heimdal
3 upgraded, 30 newly installed, 0 to remove and 481 not upgraded.
Need to get 2,877 kB/20.7 MB of archives.
After this operation, 60.8 MB of additional disk space will be used.
Do you want to continue [Y/n]? y
Err http://us.archive.ubuntu.com/ubuntu/ precise/main libevolution amd64 3.2.3-0ubuntu5
  404  Not Found [IP: 126.96.36.199 80]

...Truncated because of the dumb link limit as I'm a new user...

Failed to fetch http://us.archive.ubuntu.com/ubuntu/pool/main/e/evolution/evolution-plugins_3.2.3-0ubuntu5_amd64.deb  404  Not Found [IP: 18.104.22.168 80]
E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing?

Comment: You can use http://pastebin.com for long outputs. They are much more readable there then inside questions.
Answer: Have you tried as suggested in error message?
sudo apt-get update --fix-missing

Evolution Install


What is the difference (if any) between transition systems and finite automata?

Question: is there any difference between transition systems and finite automata? Is it that transition systems consist of both NFA (nondeterministic finite automata)
and DFA (deterministic finite automata)?
Answer: Yes, did you try wikipedia?
To quote the second paragraph [in transition systems]:

The set of states is not necessarily finite, or even countable.
The set of transitions is not necessarily finite, or even countable.
No "start" state or "final" states are given.
Answer: The difference is mainly that automata are a model of machines/programs/.. while transition systems are a model of machine/program/..'s behaviour. So one does not have the same questions and requirements for them.
You are right in thinking that they look the same: some version of directed graphs decorated in some way. However, what is important in mathematics (as in object-oriented programming) is not the objects themselves but the operations they support.
Imagine asking what is the difference between points in $\mathbb{R}^3$ and vectors in $\mathbb{R}^3$? They both are triples of numbers, no? Well, yes but this is not the point. Vectors are displacements so it is meaningful to add them, scale them. Points cannot be added meaningfully. They can be subtracted, as in $B - A$. BTW, this gives you a vector, hence $B-A$ is usually denoted $\vec{AB}$.
Answer: What you probably mean by transition system is a Kripke structure
Basically you can call any directed graph with labeled vertices a transition system.
But finite automaton is a concept and includes any language deciding machine with finite memory.
DFA and NFA are specified kinds of finite automaton and they can decide all the languages decidable by any other kind of finite automaton.
DFA and NFA are like transition systems with additional constraints:
In both cases:
 - Edges should be labeled by the members of the alphabet
 - A start state should be specified
 - A set of final states should be picked
In DFA only
 - Each state should have exactly one outward transition for every member of the alphabet
Answer: Another note, in transition systems the state is often a set of variables (atomic propositions, boolean vectors, etc.). Whereas in automata the state is almost always a label (a simple identifier such as a name or an integer or an id type of thing). The state doesn't contain anything in automata, whereas it does in transition systems.


Is there a way to bind items to the number keys in Skyrim?

Question: I'm less than impressed with the favorites menu for quick swapping spells in the middle of a fight. Is there anyway I can bind them to the number pad on the keyboard?
Comment: The way ive been playing spells can only go into your right hand (sword hand) if you have the perk that corresponds to the spells skill level. Bound sword is interesting as the perk is second level and you can dual wield them, however upon unlocking the perk Conjuration Dual Casting this is no longer the case, not sure if its a bug but bound sword will refuse to be equipped in both hands. I use Bound Sword all the time, my level 24 Breton has no weapons and uses only bound, I have yet to find a one handed weapon that does more damage.
Answer:
Press Q to bring up the quick menu. Then, hold down a number key (1 trough 8)
  while hovering over a item or spell for a second. Swap between items during
  gameplay by tapping the assigned number key.

Courtesy of Xenox's answer.
Comment: @Fabian It seems to default to weapons going into the right hand, and shields/spells going into the left hand. I haven't found a way around this, except in the case of switching from a 2H item. If you press the key the 2H is bound to while wielding the 2H, you'll go back to your previous dual-wield setup.
Comment: Oh thank Akatosh, having no quick keys was driving me around the bend. A spell-casting archer is still possible!
Comment: Could you expand on how you choose which hand the item or spell goes to when using the number buttons?
Comment: Is there a way to (maybe programmatically, with macros or aliases?) bind one key into multiple actions? For example, equip A in left hand and B in right hand with one key?
Comment: Apparently you can bind to the left hand with Shift + .
Comment: @Seven I'm not able to reproduce that behavior.
Answer: Woot.
Google never ceases to amaze me.
After reading this I figured out how to get a somewhat desired effect.
Pressing the key twice will essentially put it in both hands.
Such as bound sword, first in Left Hand for me (Right Mouse, I might end up reversing the mouse binds -_-) and then in Right Hand (Left Mouse).
So I can dual wield Bound Swords by pressing the key twice.
So far I have figured out the following:
Physical things go into your Right Hand first (Left Mouse), Magical things go into your Left Hand first (Right Mouse).
So if you bind Bound Sword to 1 and some physical sword to 2.
Pressing 1,2 with nothing equipped as a weapon will put the bound sword in your Left Hand (RM) and the Physical Sword in your Right Hand (LM).
If you press 1 twice, this will put Bound Sword in both hands.
I don't have 2 physical swords atm so I can't clarify if binding another one to 3, and then pressing 2,3 will cause you to dual wield them.
But feel free to experiment on your own!
and thank you for answering my question -:D
Comment: +1 I didn't realize that you could double-tap to dual wield a weapon. Also, welcome to the site.
Answer: If you've got the SkyUI mod installed you'll need to first disable the mod before launching the game. Then follow the directions by favoring items, spells, potions, etc and then pressing the number key you want to assign (1: bow, 2: sword, 3: shield, etc). Save (preferably a new save), exit the game, reenable SkyUI and continue from the last save. Now you'll still be able to thankfully use the number keys 1, 2, 3, etc without having to go either in to inventory or the favorites menu. Unless they create a fix you'll have to keep exiting the game and repeating this each time you change your favorites or perhaps there is a more refined GUI mod for the menus? The main thing is at least you won't be asking your enemies to wait half a minute in the middle of combat while you whip out a healing potion.


How to send private messages?

Question: How do I send private messages to another user on the website? More specifically, how can I send private messages to a moderator on the website?
Comment: A relevant response to your first more general question can be found [in this post](http://meta.math.stackexchange.com/questions/3129/how-to-contact-a-specific-user-is-there-any-sort-of-pm-private-message-fa). Regarding your more specific second question, I have seen people successfully get the attention of a moderator through tactful effort in the main chat room, but that is certainly not private.
Comment: To expand on J.W.Perry's suggestion, now there exists also a chatroom created specifically for issues needing moderator's attention, see [here](http://meta.math.stackexchange.com/a/19316). It is called [Math Mods' Office](http://chat.stackexchange.com/rooms/info/20352/math-mods-office).
Answer: If you want to contact a specific moderator: check their profile. Some moderators provide their email address in the profile, and indicate it may be used for moderation-related inquiries. If they don't, then this is not an option.
If you want to contact some moderator, you can use a free-form flag on any relevant post (or, if no such post exists, on your own post). An alternative is the moderators chatroom for maths.SE.

SE does not provide any private messaging system for contacting  non-moderators.

How to send private messages?


What will succeed the Arecibo Observatory?

Question: Just a few minutes ago, I got a notification from Space.com stating that the Arecibo Observatory will be, sadly, decommissioned due to extensive damage to its structure. So, with the loss of one of the world's largest telescopes, we will be put at a setback in space exploration and astronomy. What will succeed Arecibo and when will this happen?
Comment: If there's a website also reporting this news which doesn't put advertisements overtop of text, that would be good! Can I suggest https://www.bbc.com/news/science-environment-55008567
Comment: Related: https://astronomy.stackexchange.com/questions/40260/building-a-floating-ocean-going-giant-radio-telescope I can dream, can I?
Answer: There's no simple answer. In the immediate future, different radio telescopes around the world will pick up the slack in various ways; how that happens will depend on the needs of individual observers and collaborations. Unless someone was to build an identical observatory at the same latitude as Arecibo, with the same frequency range, receiver options and field of view . . . we'll have to spread out the observing. Given that we didn't really expect to lose Arecibo for more than about a year$^{\dagger}$, there are a lot of things up in the air. Insert "probably" and "maybe" into this answer wherever you please.
I do pulsar timing, so I'll talk about what a future without Arecibo might look like from our perspective. My collaboration's two main instruments were Arecibo and the 100-meter Green Bank Telescope (GBT), with roughly half of the observing time at Arecibo and half at the GBT. There aren't firm plans in place for the exact strategy we'll take going forward, but we'll have to decrease our observing cadence at the GBT (i.e. how often we observe certain pulsars) so we can use some of the time to observe pulsars we'd normally observe using Arecibo. So the GBT can take some of the load, but as we only have a finite amount of observing time, that impacts our other observations. We'll likely have to decrease the cadence of observations and the number of pulsars we monitor regularly.
Pulsar timing arrays will also have options with other existing instruments in the immediate future. A key one is the Canadian Hydrogen Intensity Mapping Experiment (CHIME) in Canada. The folks at CHIME will actually be able to observe all of our northern hemisphere pulsars (CHIME Pulsar Collaboration, 2017), which is excellent. In the short-term, CHIME will slightly lessen the burden of losing Arecibo. We've also done a little bit of timing with the Very Large Array, although I don't think that'll be a major player in the future.
That brings us to telescopes currently in the planning stages or under construction. The DSA-2000 (Hallinan et al. 2019) will hopefully see first light in the late 2020s and would likely become a major part of our ongoing observations. The Square Kilometer Array will also be extremely important for radio astronomy as a whole - I don't know whether we in particular will be using it a lot. The SKA should also reach completion in the next decade. And, of course, the Next Generation Very Large Array (ngVLA) will open exciting opportunities for a lot of subfields of radio astronomy.
These are just the current potential options from my corner of the astronomical world - and they're quite tentative. Arecibo was our most-used screwdriver in the toolbox, so we've certainly been hit. We'll hopefully be able to somewhat supplement the GBT with CHIME and ideally DSA-2000 a decade from now - so in that sense, those will effectively succeed Arecibo for our observations. In reality, all that means is that we won't lose quite as much of our observing capacity - but it doesn't come close to making up for the loss. Other collaborations and astronomers will presumably increase their usage at the observatories that are best equipped for their observations, depending on frontend/backend requirements and field of view.
It's not going to be fun. We'll be set back. But science will go on. We'll figure it out.

It's worth updating this answer now that the white paper for the proposed Next Generation Arecibo Telescope (NGAT) has been released. The Arecibo Observatory has proposed a phased array of small dishes mounted on a circular platform ~300 meters in diameter (or potentially multiple smaller platforms) which would be angled in various directions to provide reasonable sky coverage. One variant involved 1,112 9-meter dishes, as smaller dishes provide a higher packing efficiency and therefore maximize collecting area. Regardless of the precise configuration, the sinkhole that housed the former 305-meter dish would ideally be the new array's home.
The aim is for frequency coverage from 200 MHz to 30 GHz via four broadband receivers and a frequency-dependent field of view ranging from 6 degrees at 300 MHz to 3.5 arcminutes at 30 GHz, a significant field of view increase over its predecessor by a factor of 500. The dishes would also be equipped with transmitters for radar observations.
Whether the NGAT will be built is a question for which nobody has any answers just yet. It would cost roughly $450 million, which could take money away from other projects unless it received special Congressional funding or support from private sources. Additionally, the setup has not been tested on such a large scale, so there are some technical and engineering concerns (the group did consider but reject single-dish and classical array designs). All that said, the design was put together in an impressively short amount of time, and even if the NGAT isn't build, the scientific considerations that went into it may inspire future designs for replacement telescopes at Arecibo.

$^{\dagger}$ This isn't totally true - a scenario where we lose Arecibo had already been studied in detail. But . . . I think a lot of folks felt that it was more likely that it would be out of commission for a bit but could be salvaged in some way, and hopefully brought back online before too long. This was not the Thursday we were expecting.

Update, July 2023
Over two and a half years have passed since the announcement of the telescope's decommissioning and its subsequent unexpected collapse, meaning we can switch the question from what will succeed the Arecibo Observatory to what is succeeding the Arecibo Observatory. Some of my predictions came true, but some new developments have taken place.
The site
In October 2022, NSF announced that it would be transforming Arecibo into a new educational center, soliciting proposals for specifics. The decision did not include funds for continuing the other still-operational instruments at the site, notably, the 12-meter radio telescope and the LIDAR facility. This was effectively a death knell for the construction of the NGAT -- or alternative designs -- at Arecibo, and in fact any future science there.
Astronomers at the observatory have continued to push for a replacement at the site, however. In May of 2023, a group proposed a modified version of the NGAT, dubbed NGAT-130. NGAT-130 somewhat resembles NGAT, but consists of 102 13-meter dishes, with a collecting area that of a single 130-meter-diameter telescope. The group describes the design as "scientifically competitive" with NGAT but reasonably cheaper.
As with NGAT, and particular in light of the NSF announcement, NGAT-130 doesn't have great chances. It's entirely possible that it could be built at another site -- there are better locations for radio astronomy than Puerto Rico -- but any such plan would be strongly dependent on finding a source of funding. I'm not sure I see a way for that to happen without private funding or an (unfortunate) decision to axe DSA-2000.
Observations
I'm again writing this from my particular corner of radio astronomy -- pulsars and FRBs -- and in the intervening period since the loss of Arecibo, we've seen some alternative options emerge. Three telescopes folks have been submitting proposals to even more than previously have been the GBT, the Giant Metrewave Radio Telescope (GMRT) in India, and the Five-hundred-meter Aperture Spherical Telescope (FAST)$^{\dagger\dagger}$ in China. All three have the advantages that 1) they're world-class facilities and 2) they can cover much of the same declination range that Arecibo did. All are highly oversubscribed already, but we can still in principle see many of the objects we used to with Arecibo.
That said, in the bands I'm interested in, only FAST can reach Arecibo's sensitivity; I recently wrote in a proposal that it's currently the only telescope in the world that can see a couple of the sources I'm interested in. There are ways to mitigate this very specific issue for the GBT and GMRT: since pulsars are broadband sources with power-law spectra scaling with frequency as something like $\nu^{-2}$, you can shift observations of dim pulsars to lower frequencies. For instance, pulsars previously barely detectable at L-band (center at ~1400 MHz) with Arecibo could be observed with the GBT's 800 MHz prime focus receiver. You still won't recover the old signal-to-noise ratio, and shifting frequencies presents new issues, but hey, it's something. And after all, for most pulsars, this isn't a huge issue.
When I originally composed this answer in November 2020, I wrote that science would go on. And it has, and it will continue to do so. Arecibo is missed. But we keep going. It's what we do.

$^{\dagger\dagger}$ FAST, by the way, has turned out to be an absolutely bonkers tool for pulsar and FRB studies.
Comment: The SCIENCE must go on...
Comment: How about [LOFAR](https://en.wikipedia.org/wiki/Low-Frequency_Array_(LOFAR))?
Comment: And, in another 5 years or so, SKA.
Comment: It's time for Bezos to finance a large dish on the far side of the Moon.
Answer: As you said, the loss of Arecibo will definitely put a dent in the field of radio astronomy. As for what will help take its place - there are a couple options.
Green Bank Observatory has been and still is quite a widely-used radio observatory. It helps in many initiatives, not limited to but including Breakthrough Listen. I know there are many people who work at/with Arecibo who are connected in various ways to Green Bank, so I imagine some of the observational responsibilities/tasks or projects might end up in the hands of Green Bank.
In addition, the Allen Telescope Array is currently undergoing refurbishment to be used on a larger scale for observing various radio sources - eg. FRB's. I would think that in the near future, when things get up to scale and ready for use, the ATA would also help fill in the void that the loss of Arecibo has created.
I think it's also worth mentioning FAST - China's Five Hundred Meter Aperture Spherical Telescope. The name is a bit misleading, since the aperture that the telescope uses isn't 500 meters in diameter, but it has been discovering pulsars at a pretty fast (pun intended) rate over the past few years. You can see their discoveries here.
And, as I'm sure many of you in the radio astronomy world have heard, the Square Kilometer Array is projected to have its first light in 2027. It will be the largest and most powerful/sensitive radio telescope ever created when it is finished, and will most certainly be invaluable to the field of radio astronomy.
Answer: The big loss is to radar astronomy.  Arecibo was one of only two radar telescopes in the world in regular use, and was by far the more powerful: a 300 meter antenna and megawatt transmitter, versus Goldstone's 70-meter antenna and 500-kilowatt transmitter.  I'm not aware of any plans for successors: FAST can't be fitted with a transmitter without a complete reconstruction of the receiver platform, and none of the other dishes even comes close to Arecibo's size.
Comment: This is slightly related, in as much as it references its available transmit power: [Has DSS-43 ever been used in high power mode (>>20 kW) for an emergency situation?](https://space.stackexchange.com/q/18956/12102) you are welcome to add an answer there explaining that the high power capability is not there only for emergency spacecraft communications, but is used regularly for radar astronomy. Until recent days that had never occurred to me. I think you answer will be well-received (pun intended).
Comment: @Mark I added a link to the Goldstone wiki page which I assume might be helpful.
Answer:
What will succeed the Arecibo Observatory?

(As already mentioned in this answer) One of Arecibo's uses was as a powerful transmitter for radar astronomy. For more on that see

this answer to Is there any role today that would justify building a large single dish radio telescope to replace Arecibo?
Farthest distance to a solar system object that's been measured by radar?
How did Arecibo detect methane lakes on Titan, and image Saturn's rings?
Why is Saturn invisible in this radar image of its rings?
Why was the 100m Green Bank dish needed together with DSN's 70m Goldstone dish to detect Chandrayaan-1 in lunar orbit? (references therein)

Well it seems that the iconic Green Bank (Radio) Telescope (website) has now been fitted with a transmitter in order to beam powerful, well collimated radar pulses into space for the purposes of radar astronomy!
Per this press release:

GBO's Green Bank Telescope (GBT) — the world's largest fully steerable radio telescope — was outfitted with a new transmitter developed by Raytheon Intelligence & Space, allowing it to transmit a radar signal into space. The NRAO's continent-wide Very Long Baseline Array (VLBA) received the reflected signal and produced images of the Apollo 15 moon landing site.
The proof-of-concept test, culminating a two-year effort, paves the way for designing a more powerful transmitter for the telescope. More power will allow enhanced detection and imaging of small objects passing by the Earth, moons orbiting around other planets and other debris in the Solar System.
The technology was developed as part of a cooperative research and development agreement between NRAO, GBO, and Raytheon.
"This project opens a whole new range of capabilities for both NRAO and GBO," said Tony Beasley, director of the National Radio Astronomy Observatory and vice president for Radio Astronomy at Associated Universities, Inc. (AUI). "We've participated before in important radar studies of the Solar System, but turning the GBT into a steerable planetary radar transmitter will greatly expand our ability to pursue intriguing new lines of research."
Using the information collected with this latest test, the participants will finalize a plan to develop a 500-kilowatt, high-power radar system that can image objects in the Solar System with unprecedented detail and sensitivity. The increased performance also will allow astronomers to use radar signals as far away as the orbits of Uranus and Neptune, increasing our understanding of the Solar System.
"The planned system will be a leap forward in radar science, allowing access to never before seen features of the Solar System from right here on Earth," said Karen O'Neil, the Green Bank Observatory site director.

From NRAO's Successful Test Paves Way for New Planetary Radar (looks a bit astigmatic)
caption:

GBT-VLBA radar image of the region where Apollo 15 landed in 1971. The snake-like feature is Hadley Rille, a remnant of ancient volcanic activity, probably a collapsed lava tube. The crater at top, alongside the rille, is called Hadley C and is about 6 kilometers in diameter. This image shows objects as small as 5 meters across.

Credit: NRAO/GBO/Raytheon/NSF/AUI
Comment: @B--rian thanks for the edit, found one more