Properties of Water

Question: Water is a substance characterised by properties such as high boiling point and its ability to act as a universal solvent. So why do we say that "the smallest unit of water is a water molecule"? Single molecules cannot possess such properties. Am I missing something here?
Comment: You need *a lot* more water molecules until you have water. A drop of 10,000 will show a huge difference e.g. in vapour pressure. Even a billion would still be quite strongly influenced by it's surface, and not show the ideal bulk water properties.
Comment: Not all water is defined the same. "Water" defines a molecule - $H_2O$ - but in our everyday parlance also a bulk substance which is a collection of such molecules. Perhaps it would help to say "the smallest unit of a volume of water molecules is a single water molecule". Not very enlightening perhaps.
Comment: @J.Smith I don't think that's going to solve your question, because you still need to define "properties". The properties in question can depend on how many water molecules you have.
Comment: The problem here is that conventional usage of "the smallest unit of water is a water molecule" leaves out a lot of qualifications. The sentence ignores the qualification that some properties are intrinsic to the molecule but others only appear when there are a lot of molecules together. Every molecule in bulk water has the same mass, geometry and electric dipole but many of the bulk properties only appear when there are millions of the molecules (since they depend on how the molecules interact).
Comment: Water is a substance. Would it perhaps be more correct to say that a water molecule is the smallest unit that defines the properties of the substance water?
Answer: No, you aren't missing anything. Indeed, one molecule doesn't have any boiling point, neither high nor low. In fact, you've just stumbled upon one of the holes deliberately left in the early definitions. It is a very common thing in the textbooks on natural sciences: first we give the kids a simplified definition, so they don't have to read a ton of prerequisites before they start reading the book, and then they kinda already know what a molecule is, so why bother returning to it.
If we were to invent a more-or-less realistic definition of a molecule in simple terms, it would go like this: "the smallest unit of matter such that the matter retains its properties when split to these units and then allowed to settle down".
Indeed, say we have a glass of water, and we magically split it to individual molecules and then step aside. The molecules will assemble back into liquid water; probably it will be all over the place, but who cares, it is liquid water all the same, with unchanged boiling point and stuff.
Now what if we magically split a sample of some matter to smaller units, like atoms? Well, as soon as we step aside, those will assemble back to molecules, but not necessarily the same molecules. They may well be different, and so will be the physical properties of the sample.
This definition is not without its flaws either, but that's another story.
Comment: Many molecules won't survive being taken into vacuum. ;-)  Imo trying to define real world stuff like molecules in terms of abstract entities like "smallest unit of matter" is generally a bit of a philosophical exercise in futility. You get to statements which are often flawed, true in best case, but all pretty useless in gaining further understanding.
Answer: The word "water" can be used to mean different things. In everyday language we usually use "water" to mean a large collection of water molecules, which we associate with such phenomena as boiling, freezing, solubilization of salts, etc. We can conjure a different definition of "water", namely, a single water molecule, with molecular formula $\ce{H2O}$. These are not the same thing. In a way, one is a subset of the other.
The solvation capabilities and boiling point of water are thermodynamic properties, usually (but not only) defined for macroscopic ("bulk") quantities of water under equilibrium conditions. The boiling point defines a temperature at which a transition occurs between the liquid and the gas states of a substance. The boiling point is a property of the ensemble, the entire collection of molecules. It is important to emphasize that when chemists discuss properties such as boiling points they are pretty strict. Such properties depend on the pressure, amongst other things. Even the geometry of the water volume can influence the experimentally observed boiling point. Chemical thermodynamic properties belong by definition to a collection of molecules under well defined equilibrium (constant) conditions. Thermodynamicists observed that properties like solubility and phase transition temperatures reach limiting values provided you have sufficient molecules, that is, above a limit dealing with bulk matter. In fact, a thermodynamic limit can be defined by observing how properties change with the amount of a substance.
It doesn't make much sense to attribute a bulk property such as a boiling point to a single water molecule, but you can say that a single molecule of water has intrinsic molecular properties that lead reproducibly, when the molecule aggregates with others of its kind, to specific macroscopic properties.
On a historical note, properties such as the boiling point were defined quantitatively as the foundations of thermodynamics were layed down and a formal molecular theory of matter began to be developed, the latter involving amongst other things statement of Avogadro's hypothesis and, later, advances in spectroscopy.
Comment: Thanks for the response Try Hard. Does that mean that we cannot define a single molecule of water to be the substance "water"? Since water is characterised by the aforementioned thermodynamic properties.
Comment: Water is divisible down to the molecular level. It exhibits properties (thermodynamic or possibly other) that may depend on the number of such molecules and their interactions.
Comment: Then let me ask this question: What properties characterise a substance? Are they thermodynamic properties or something else? What am I missing here?
Comment: The word "substance" typically implies a macroscopic ensemble, a large collection of molecules. A thermodynamic description might be adequate, assuming the substance is in equilibrium, which for a pure homogeneous liquid such as water would mean stating what temperature and pressure it is at. Then you can proceed to list all of the properties you might be interested, extensive (such as volume, enthalpy of formation, etc) or intensive (molar properties such as molar heat capacities, thermal expansion coefficient, etc).
Comment: With thermodynamic properties you would not be speaking of one water molecule, but of the properties of the ensemble of molecules. But of course properties can also include spectroscopic properties that are impossible to explain without an atomic/molecular/QM description of matter. When you start dealing with certain spectroscopic properties, you move away from the aggregate properties and shift into the realm of the molecule. The aggregate properties become "perturbations" on molecular properties that are similar in an ensemble and in a single molecule. Often, but not always.
Answer:
the smallest unit of water is a water molecule

To say that a molecule is the smallest unit of a substance that shares its properties is not a good definition, and has the same problems as defining an atom as the smallest unit that has all the properties of a given element.
For the definition of an atom, you can salvage that definition by specifying exactly what properties are retained. For example, the electronic transitions involving inner electrons will be the same for an isolate atom and for different allotropes of the element (and you can use that to identify atoms from their spectrum even if they are extraterrestrial).
Similarly, you can specify a small set of properties that a bulk sample of a pure molecular substance has in common with a single one of the molecules it contains: Composition, bond lengths and angles (with some minor variations), electronic spectra (more or less).
Of course, if your bulk sample is in the gas phase, there will be more properties that are the same, so pure water in the gas phase or water in air will share most properties with a single isolated water molecule.

Properties of Water


What is the point of Legendary Magician of Dark's effect?

Question: Legendary Magician of Dark says that if it attacks reduces the opponent's life points to 0 and some other conditions are met, you win the match. But wouldn't you win anyway just by having reduced their LP to zero? What's the point?
Answer: The key difference here is that a duel consists of a single game and also a match consists of a best-of-three duels game. Official Tournaments consist of rounds of matches.
So, with this card if you manage to reduce your opponent's LP to 0 and fulfill the other conditions you win the best-of-three games immediately.
Be aware that Legendary Magician of Dark is one of the Tournament Prize Cards that are awarded to World Tournament Champions. All these have the common trait of making you win the match.
Unfortunately, this cards cannot be used in a oficial tournament; you can only use them on friendly and traditional games, that is, if you have won one of course :)
Comment: @Thunderforge yes, surely this completely ruined the whole idea of Tournaments consisting of matches, not to mention being discouraging for some players, as well as giving more advantage to former Champions.
Comment: And the reason that the are forbidden in official tournaments is that players were surrendering before the monster could attack, this only losing one duel instead of the whole match. This was seen as bad sportsmanship, and there wasn't really a way to modify the surrendering rules effectively to prevent this, so they banned them.
Answer: You win a duel by reducing your opponents life to 0 and you win the match by winning 2 duels. So what this means is you can win the match in 1 duel instead of 2.
Wiki Link

The first Duelist to win two Duels is declared the winner of the Match. If a Duelist loses two Duels, they are considered to have lost the Match. If both Duelists have one loss in a Match and both receive a simultaneous Game Loss penalty, the Match is considered a Double Loss
Answer: Reducing your opponents life points to 0 is one of the ways that wins you a duel, a match is the best 2 out of 3 duels. The Match Winners cause you to win the entire match in one duel, once their conditions are met.
The first match winner was Victory Dragon, sent out as a promotional card in Shonen Jump magazine. When it came into play, right before you would lose the duel to it's effect people would simply concede the duel, to have a chance to win duels 2 and 3 and still win the match. This caused Victory Dragon to be forbidden very quickly for tournament play.
After Victory Dragon, all match winners have been prizes at high level tournaments, and all match winners have the same forbidden status, they are more showpieces than usable cards. In fact, after that first one, they all have forbidden status printed on the card, just below the text box on the left: "This card cannot be used in a duel."


Why are side mirrors glass?

Question: Why are side mirrors glass instead of plastic? As far as I can tell they are just glass with a layer of reflective material adhered to the surface, would plastic not be a better alternative as it would not shatter if it fell out.
Comment: Welcome to the site. Why would the glass fall out in the first place? They are captured within the sideviews. They use glass because glass is ***very*** hard and can stand to be cleaned very often, which plastic wears out in a short period of time in relative terms (plastic scratches easily in comparison).
Comment: @Pᴀᴜʟsᴛᴇʀ2 Thanks, I asked because mine was recently knocked out by a truck driving past me, it just occurred to me that a treated plastic may have a better time surviving the impact. Although I take your point about glass being far more resistive to cleaning and wear.
Comment: If it fell out due to a truck going past, there was an issue with it in the first place. It's just something you don't regularly see. I've replaced a couple of side view mirrors and they are really a PITB to get done (taking out more than putting back).
Comment: @Pᴀᴜʟsᴛᴇʀ2 Apologies I should have been more clear in my initial explanation the truck hit the wing-mirror causing the glass to fall out and smash
Comment: No apologies needed ... physical contact causes all kinds of bad things to happen!
Answer: The first and simplest reason is that glass is cheap cf. a plastic that will support the uv rays that will hit the mirror during use. Then, the glass is designed to shatter into small pieces when it gets broken not forming shards which are more likely to cause injury.
Some cars have plastic headlights and we now have to buy products to reduce the discolouration due to aging....

Why are side mirrors glass?


Trail maintenance: what is considered inappropriate when maintaining a trail?

Question: Every year at the beginning of the spring, I do preliminary hikes and do some light maintenance (removing broken tree from the trail etc...) but this question made me realize to what extend should I maintain the trail? And if some actions could be considered as inappropriate?
Comment: It largely depends on who owns the trail.
Comment: In the US at the present time, a lot of government agencies that used to do trail maintenance have simply given up due to lack of funding and staff. This started in 2008 with the economic crisis, but it's starting to look like the new normal. A couple of areas seeing essentially no maintenance are Angeles National Forest and Ventana National Wilderness. In these areas, it may be reasonable just to go ahead and do something that you think needs doing. For example, the last time I was in in Ventana NW, there were trails completely blocked by poison oak. Nobody will complain if you remove some.
Answer: It's always OK to move fallen debris from the trail, assuming you are sure you are on a real official trail.  Make sure you don't accidentally remove "brushed in" trail entrances.  That is where brush was deliberately piled so that a trail is not used, hopefully eventually reverting to just woods again.  If you're on anything with clearly deliberate blazes, you're safe.
For anything more than clearing debris without tools, you need to ask the owner or administrating agency.  Public agencies usually have rules that they don't want any unauthorized trail work.  This is just to make sure they don't get morons doing stupid things and to avoid liability.  If you get to know the manager, forester, committee, or whatever that oversees a particularly area and they get to know and trust you, you usually end up with a informal arrangement where they trust you doing trail work without asking first and without them formally giving permission.
Some groups can be real control freaks and don't want anyone, even those that clearly know what they are doing, doing trail work without 3 levels of approval.  I know one organization that takes this to the extreme.  They have a "steward" assigned to each property, and then a few master stewards for groups of properties.  Even though I actually built some of their trails, I can't go out there when I have a spare hour and clean up a trail without prior approval.  I don't feel I should work that hard to give away my time for free, so I find a different place to do trail work.  There are plenty of places run by other groups or organizations that are happy to have me help.
I'm on the trails committee in my town, so I am actually authorized to go out with a chain saw and clean up a mess on any of the town land whenever I see a need.  However, we don't want people going so far as using a chain saw without talking to us first, but even then there are certain individuals and groups we know aren't going to make a mess and nobody is going to make a fuss about one of them going out and clearing a major blow-down.  But, we're not going to tell you that if you come and ask as a individual, especially if we don't know you.
A good way to get known is to volunteer for organized trail work.  I don't know how other towns work but we probably have 1-3 formally organized events a month except when the snow it too deep to do much useful.  After you show up a few times, the organizers will recognize you and you'll be trusted a lot more.  You might even mention a particular trail you think needs some work and offer to take care of it over the next week or two.  Once you're known, you'll probably get a quiet unofficial nod and it works out for everyone.
Added:
A good example illustrating the issues envolved came up last night at the town Trails Committe (of which I am a appointed member) meeting.  We learned that someone or some group has gone out to several trails with a leaf blower and "cleaned" them off down to bare ground.  We're pretty sure it was the local mountain bikers club, or at least individuals from there.  We have a generally good relationship with the mountain bikers, and they show up regularly to help out with scheduled official trail work.  We don't really want to get in anyone's face about this, but would rather the leaf blowing wasn't happening, at least not without prior approval.  The problem is that this promotes erosion, and also gives the trail a unnatural and less wilderness feel.  We also really don't like anyone working on trails with powered tools without coordinating with us first.
It's too soon to tell how this will be resolved.  Some polite off the record conversations will be had with those envolved, and that will probably resolve this issue.  This is a small town, we pretty much know who is doing what, and we also understand they probably thought they were improving things.  Mountain bikers don't like downed leaves because they can be slippery and cover up hazards.
In any case, this illustrates some of the problems you may create for other people when doing what you think is helpful trail work.  I'd say catagorically to never go so far as using powered tools (chainsaw, weed whacker, brush mower, leaf blower, etc) without getting the proper permission first.  Picking up trail debris and even pruning back small overhanging limbs with a hand clipper would be OK in most cases.  Everything in between is a gray area where you should first at least know who the owners or managers are, what rules there might be, etc.
Comment: This was a wonderful answer. I also want to add that I have had a lot of success just asking a park ranger (say, at Yosemite) if they wanted me to bring some light tools to help clear the trail as I went along.
Answer: If you want to help maintain trails, its usually better to volunteer with an existing organization with a relationship with the park, than to try and act on your own.  Its not always going to be possible to know what the exact appropriate / inappropriate line is when you're out on your own hiking.
Sometimes park services have conflicting priorities (for instance, balancing the needs of hikers with mandates to protect the natural environment, or balancing the needs of hikers with rock climbers or mountain bikers).  Individuals that perform large, impactful "improvement" projects can aggrevate tensions within the park in a way that planned volunteer events don't.
Removing broken treebranches from a trail by hand is probably fine, but much more than that (anything that would require tools, building systems of cairns, etc) should probably be left for formal trail improvements.
Comment: Volunteering is the way to go, I totally forgot that.
Answer: As a general rule in the US

Private:  Ask the landowner, they are probably happy for the help.
State:  Don't touch anything.
National Forest:  Do it if it's an established trail.  It's probably not "approved" but the rangers and other hikers will appreciate it.
National Park:  Don't touch anything.  Also avoid looking too hard if you can.
Comment: +1 for "avoid looking too hard".
Park services have strange and sometimes conficting priorities (balancing needs of hikers with nature conservancy / biodiversity, for instance, or balancing the needs of regular hikers with rock climbers).  If one person starts making large, impactful changes on a trail on his own (setting up cairns, sawing off branches that have fallen in the trail) it could aggrevate existing tensions in the park
Comment: I'm not getting what you are trying to convey by not looking too hard in a national park.
Comment: -1 for "do it if it's an established trail", since both "it" and "established trail" are too vague; also for over-generalizing "state" since there are 50 of them with widely varying policies on public land; also for excessively snarky framing of NPS issues.


How can an OpenGL visualization run for weeks?

Question: For an art project, I need to make an OpenGL visualization run for at least 4 weeks. Is this possible? I can pretty much choose the rest of the hardware and software.
Answer: Opengl doesn't care how long it has been running.
However there are a few other challenges with long running applications in general:

memory leaks, both normal ones and opengl resources.
lack of precision after incrementing a float to a big number. After 4 weeks your millisecond counter will be at 2.4 billion which is well beyond the point where integers cannot be expressed exactly in a single precision floating point number. Make sure that those numbers are wrapped down to a smaller value explicitly.
Answer: Yes, of course. The duration it has to run for doesn't make much difference. Your Android smartphone's main GUI is OpenGL, and phones typically have an uptime longer than four weeks. Air Traffic Control centres might use OpenGL SC (the subset of GL for safety-critical applications) continuously for months or years.
If you do have free choice of software, OpenGL is probably not the thing you're concerned about. Artists often use Processing for art projects, because it's simple and lets you get to drawing stuff on the screen with minimal 'boilerplate' code. It's very nice for drawing maths-as-art: particle systems, L-systems, strange attractors, &c. For a 3D scene that's supposed to look like the real world, Unity 3D might be a good choice: it's a game engine that handles all the use of OpenGL or DirectX for you, so you can create the scene you want to draw without much programming.
If you have more recommendation- or opinion-based questions about what might be useful to look at, maybe drop into our chat room and be ready to tell us what kind of art project you're making.
Answer: To make an application, OpenGL or not, run for extended time safety-critical applications are mostly built to restart on error, be it a driver bug, your application is broken, or hardware failure. If function during the downtime is required you need redundnacy. In your case maybe start multiple sessions of the same program, or even having dual computers running the same program outputing to the same display.
OpenGL is no different from other applications, BUT vendors optimize and test for certain kinds of customers. Your scenario is not common so be suspicious.


How plausible is a photosynthetic parasite that targets animal hosts?

Question: Real parasitic plants target only plant hosts and in certain species fungal hosts. How plausible are parasitic plants (or algae, lichen, or other photosynthetic organisms) that targets animal hosts similarly to parasitic fungi? What would be efficient survival strategies? What evolutionary pressures would select for animal hosts?
Comment: Somewhat similar to [this](https://worldbuilding.stackexchange.com/questions/15263/how-would-a-plant-evolve-to-germinate-inside-a-person-or-other-animal)
Comment: I am going to say this is not a duplicated.  This question seems to be interested in parasites on the OUTSIDE of the human, whereas the suggested duplicate focuses on plants growing inside a human, which is technically parasitism I guess but not even close to the same.  An external parasite takes energy/food from what the host has already processed.  The linked dupe on the other hand the plant would absorb from the GI tract prior to the human processing it.
Comment: Atheletes foot and warts are fungi that target humans, do they count?
Answer: I think evolution could handle this, given enough time and the right pressures.  Here are the steps, in brief:

Start with an existing carnivorous plant like a butterwort or
bladderwort.
Increase the potency of the mucilage for both its stickiness and digestive abilities
Develop topipotency, the ability to re-grow from stems that stick to a host and that break off.  Some species might already be there.
Develop an anesthetic within the mucilage or otherwise so the host doesn't try to bite or scratch at the infected site.

Before you know it, you've got one of these blooming on your elbow, drawing nutrition out of your blood and muscles.

How lovely.
Comment: Hah - the new tattoo!
Answer: It's unlikely but not impossible.
Parasites tend to lose abilities that they once had, as they evolve. For example, many plants which parasitize other plants have lost their ability to photosynthesize, and/or  the ability to grow leaves. This may be because what would have been a deleterious mutation in a non -parasite becomes harmless when it is the host that suffers any consequences.
So I'd rather expect a plant that could parasitize animals would similarly later lose the ability to photosynthesize.
BTW I once read that there are plants that prey on animals much bigger than themselves. Cacti with strong barbed spines and loosely attached "daughters". The daughters snag a passing animal. All attempts by that animal to free itself simply drive the spines deeper. The wound becomes infected. The animal dies. The baby cactus roots itself in ground well moistened and fertilised by the rotting  corpse. Not strictly speaking a parasite, but very yukky.
Roll forwards many million years and it may have evolved to feed off the animal without killing it. Roots that colonise the wound and secrete antibiotics to prevent bacterial infection, maybe?
Comment: they loose non necessary skills, because they have not way to remember them, no books, no ,,, nothing, and they have new survival goal. Live of parasites, specially on mobile hosts is tough, unpredictable and they have to be prepared(and they are, most of them is hard to kill at some stages). Although my second though was your first string, but now I would say more definite not. That cactus, yes, that's valid strategy.
Answer: The main initial benefit of animal hosts is that they are mobile and often migratory, allowing the plant life to spread to a larger area.  Many plant species use this strategy by producing fruit that is eaten by animals dispersing the seeds.  Given the numerous types of fruits this is clearly a successful strategy.  It would not be much of a shift to have the plant itself (or offshoots) hang on to the animal allowing it to spread much further.
After they learn to hang on, they could further develop with different host animals in various ways: sharing nutrients from photosynthesis in a symbiotic manner like lichens.  Becoming more carnivorous and feeding off their hosts exclusively.  They might stop rooting in the ground at any point in their cycle and live spreading from animal to animal.  Possibly even becoming wholly dependent on specific animal species for their reproduction cycle.
Comment: Parasitism might be restricted to the sporophyte, carposporophyte, tetrasporophyte or gametophyte generation depending on the ancestral organism.


How to say: How does he look like? Happy, sad?

Question: How to say the sentence from the title. I know the affirmative version: Il a l'air heureux. But the question remains evasive: Il a quel air? Quel air a-t-il? Sounds a bit rough, cannot explain why.
Comment: @SteeveDroz to which question I would be tempted to answer: "il est 2 heures et demi"
Comment: "quel air a-t-il ?" would be if there are only a given number of airs allowed or existing, something that sounds like "Which look does he have?". That's why it seems rough to you.
Comment: Being at work, I am now trying to hide my laughter. Thanks for nothing (and everything at once) @njzk2!
Answer: I would say "Comment a-t-il l'air d'aller ?" or simply "Comment va-t-il ?".
"De quoi a-t-il l'air ?" is possible too, but it refers more to the physical appearance.
Comment: +1. You can also add : "Est-ce qu'il va bien?"
Comment: Alors, une bonne situation où on peut poser cette question serait: Il a l'air de quoi ? Il a l'air grande et beau.
Mais, pour le sentiment, la question serait: Il va comment ?
Answer: Avoir l'air is a rather ambiguous expression to use for a question.  The question might also be about someone's appearance or intentions, so it's hard to tell what kind of answer you are seeking.
If you are specifically asking about a person's mood, you could ask

Quelle est son humeur?
Answer: I think most French people would say: "Il a l'air de quoi?", or more formally: "De quoi a-t-il l'air?".
This could however be interpreted as pertaining more to the physical appearance of the person. A way to ask the question that may be closer to your emaning would be the say: "Il a l'air comment?", although this is not particularly formal.
Comment: I agree that "Il a l'air de quoi?" questions more the look of the individual. It would be a translation to "**What** does he look like?"
Comment: We never say "Il a l'air de quoi?" dans le sens de "Comment a-t-il l'air d'aller?". The answer to "Il a l'air de quoi?" would be "Il a l'air d'un clown", but not "Il a l'air triste".
Answer: In French, "air" would be the right word to use, but as it refers to so many different aspects of the person (look, mood, fitness, and so on), you would have to be more specific:

Il a l'air heureux ? (Familiar)
A-t-il l'air heureux ? (Formal)

Which would translate to

Does he look happy?
Answer: I'd say "Il a l'air comment? Joyeux, triste ?"


What are the recent advancements in building a unified theory of bounded rationality?

Question: It seems that bounded rationality models focus on explaining a particular psychological bias, in a very specific way. In particular, its seems that the state of the art consensus is that one size does not fit all. The prevalence of framing effects makes this issue very difficult, but is there any way to think of a general approach to modelling bounded rationality.
Is it regret minimization, or random choice, or rational inattention?
Comment: Not a full answer or one using recent advances, but I'd summarize bounded rationality as the "optimization over optimizations." When it is costly to optimize, folks develop rules of thumb. This brings up a funny infinite regress though. Addressing that, there's a Lipman (1991) piece, "How to Decide How to Decide How to...: Modeling Limited Rationality."
Comment: Nice, comment I did not know Lipman quote.
Comment: This paper is very enlightening, Is Everything Connected, biases in Bounded Rationality http://www.columbia.edu/~po2205/papers/DeanOrtoleva_Relationship.pdf
Answer: The term bounded rationality was introduced by Herbert Simon. He wrote

"The term, bounded rationality, is used to designate rational choice
  that takes into account the cognitive limitations of both knowledge
  and cognitive capacity. Bounded rationality is a central theme in
  behavioral economics. It is concerned with the ways in which the
  actual decision-making process influences decisions."

This passage seems to focus only on incomplete information, limited information processing time, limited knowledge -and not on any "psychological biases" like framing effects and the like. But the passage is complemented by a last phrase

"Theories of bounded rationality relax one or more assumptions of
  standard expected utility theory"

which suddenly opens the content of the term to anything. So "bounded rationality" has come to signify "deviations from strict rationality in any way", making it rather impossible to arrive at a general, all-encompassing, or even most-encompassing, modeling approach, that at the same time, will be specific enough so as to be operational.
Answer: One recent paper that is being positioned as a very wide-ranging theory of bounded rationality (although certainly it doesn't come close to capturing every insight in the field) is Gabaix's forthcoming QJE, A Sparsity-Based Model of Bounded Rationality.
Gabaix formulates a fairly general model where agents can rationally decide to pay limited attention to each economic variable (depending on its importance), and derives the consequences for the classic features of consumer theory, general equilibrium, and so on. Since this is not my field, I am not completely sure how this paper differs from the preexisting literature on rational inattention, but it does seem quite general and ambitious.
Comment: nice paper, very interesting.
Answer: Herbert A. Simon coined the terms bounded rationality and satisficing. The connection between these concepts is that, since our cognitive aparatus has real-world bounds in terms of what can achieve (e.g. cognitive processing, memory, time, etc) people are often do not able to optimize. Hence they often satisfy with a sufficient good outcome (i.e. "satisfice").
The theory of expected utility still holds but people, instead of trying to maximize it, try to achieve at least a sufficient good level (threshold, aspiration level) in terms of expected utility.
Hence, the first course of action that people find which is able to satisfy (be greater than) that utility level is then chosen to be applied.
Goal-programming is one of the existing decision-theory methods which is based on Simon's underlying principle.
Comment: This answer could be greatly improved by linking to the approach, the literature and by clarifying who Simon is.
Comment: I've just included your suggestions of improvement, which I appreciate
Comment: Thank you but your strongest claim "A general approach to modelling bounded rationality is well-developed and has been proposed in the literature" is still without any source. As it stands, this answer just clarifies the basic concepts of BR.
Comment: Just changed that
Comment: Not at all. You merely linked to linked to a bio of Simon and the definition of goal-programming. How is either of these a literature of a general approach? Perhaps you misunderstand the question?


Can a Bitcoin transaction have the same address for input and output?

Question: Can a Bitcoin transaction have the same address for input and output, i.e. spend some of the coins that are linked to an address and send the rest back to that same address? Is there some reason why it is imprudent to do so?
Answer: At the protocol level, there is no such thing as "balance of an address". There are individual unspent outputs (like coins in a wallet), which must be individually spent. You can't partially spend an output, but you can split/combine it.
So for example, you have a 70 BTC output assigned to an address, and a 80 BTC output assigned to the same address. Some software will show this as 150 BTC balance of that address, but you still need to know the individual outputs to spend them. If you want to send 100 BTC to someone, you would consume those two outputs in a transaction, and create two new outputs using it, a 100 BTC one (to the recipient) and a 50 BTC one (back to yourself).
Whether you use a new address for that 50 BTC "change" output doesn't matter to the protocol. You can reuse the old one, but will make it obvious to everyone in the world which who was the sender and who was the recipient in the transaction. Using a fresh address improves privacy.
Seen otherwise: reusing the same address may look convenient, as it allows you (and software you use) to more easily distinguish what your balance is, but it does this at the cost of making it obvious to everyone in the world. This hurts Bitcoin's privacy - not just yours, but that of everyone in the system.
Answer: Yes it can. See this transaction below.
https://blockchair.com/bitcoin/transaction/b345c51064074f5d6c0e11187326567765eb4fa3236ca1be440ccb5745775238
Answer: Yes, and you might want to read up on handling change as that's one good reason to do so.
Answer: You can even have a transaction with only 1 input and 1 output, both being the same address


Are old rev*.dat file safely deletable?

Question: based on this answer rev files contains information for dealing with reorgs.
If I think reorg longer than let's say a month are highly unlikely, could I safely delete rev*.dat files older than a month?
Comment: Why not to test deleting rev*.dat files in `testnet` environment and check? :)
Answer: No. You need the rev*.dat files as much as you need the blk*.dat files. If Bitcoin Core cannot find the rev*.dat, it will need to regenerate them by reindexing the databases. Removing them effectively corrupts the databases.
If you want to save disk space, you can use pruning instead. This will have Bitcoin Core properly remove blk*.dat and rev*.dat files to save disk space.
Comment: I don't want to prune. I need a full node. Did you actually test what you are saying? In which occasion the node will recognize some rev files are missing? I am asking because I stopped the node, moved some old rev files and restarted the node. There is no error message in the log neither the regenerated files
Comment: A pruned node is still a full node. I have done this in the past and it resulted in a reindex. It looks like Bitcoin Core doesn't verify the undo data as much as it does for the block data, but it still does some verification. If you delete the wrong files, it will have problems.
Comment: yeah sorry for the wrong terminology, anyway I want all the blockchain data. I still suppose the wrong files causing problems are only the recent rev* files. I would like to see other answer
Comment: I can now confirm after testing my node started to recreate rev files in a very long process after I deleted it. So somehow are needed also the old ones

Are old rev*.dat file safely deletable?


Does this picture show an Arizona fire department extinguishing a fire in Mexico?

Question: This was posted by the Swiss news portal watson.ch:
Comment: This would be a better question if you quoted any text that described the photo, and/or linked the post.
Comment: Why is this a notable claim which invites skepticism? Seems at most like a mildly interesting claim which invites curiosity.
Comment: If they did that and googled *Arizona fire department extinguishing a fire in Mexico* they wouldn't need to ask because the first hit is Reddit. Its top, 8.6k+, comment is *This happened in 2012.* and links to the original article written by Manuel C. Coppola for Nogales International on May 10, 2012, and not just some guy named "Dave" in 2019.
Comment: Neighboring towns' fire departments often have reciprocal help agreements, even across borders.  This: https://www.cnn.com/2007/US/11/14/border.firetruck/ made the news when post-9/11 immigration checks held up a Quebec fire truck from entering the US to assist with a blaze.
Comment: The article OP mentioned is https://www.watson.ch/!556353724 which cites this reddit cross-post as a source: https://www.reddit.com/r/Mujico/comments/fuccd5/pinch%C3%A9_fuego_fugas/ and a few steps later we find this: https://www.nogalesinternational.com/news/blaze-in-nogales-sonora-battled-from-both-sides-of-the/article_3267761e-9b0d-11e1-85e7-001a4bcf887a.html
Answer: Yes. At Nogales, Arizona, in 2012, according to this story, found using Google Image Search:

Nogales, Arizona, firefighters rushed to the international border with
Mexico at Morley Avenue after a blaze broke out at the Hotel San
Enrique in Nogales, Sonora, Mexico. The hotel had been a lodging for
migrants waiting to cross illegally into the United States.
The building was empty at the time of the fire and no injuries were
reported. About 10 firetrucks responded from the Mexican side of the
fence, while Nogales Fire Chief Hector Robles dispatched the local
department's $827,000 ladder truck for its maiden trip to an actual
fire.

The Nogales, Arizona fire department have helped the Nogales, Sonora, Mexico fire department on other occasions:

December 17, 2018 at 10:08 PM MST - Updated December 17 at 10:08 PM
NOGALES, AZ (Tucson News Now) - A 20-foot high fence with mesh and
razor wire divides sister cites Nogales, Sonora, Mexico and Nogales,
AZ. It's a border, but it doesn't stop the relationship between both
fire departments along the border. A relationship that came in handy
during a fire this past weekend.
On Friday, Dec. 14 firefighters in Nogales, Arizona saw smoke. They
called it in, thinking the fire was on the American side. Crews
arrived and discovered the fire was actually on the Mexican side of
the border. There was an issue though for the firefighters in Mexico,
they didn't have water flowing to the hydrants.
The firefighters from Arizona acted fast and helped out their
counterparts across the border. They hooked up a hose to a hydrant.
Ran the hose to the fence, and there crews ran into another issue. The
mesh fence that is about 7-feet high with sharp edges. The edges could
cut the hose.
One of the captains decided to lay a ladder on the mesh fence. So the
hose would not get damaged and it worked. They tossed the hose to the
firefighters in Mexico and they were able to put water on the fire.
It took the firefighters several hours to put the fire out, but out of
the three homes that were burned, only one was destroyed.
Tucson News Now spoke with the fire Chief of the Mexican fire
department about the two departments working together. He says the two
have been helping out each other for years.
Comment: Looks good.  [Google Streetview](https://firstname.lastname@example.com,-110.9420348,3a,75y,94.55h,99.06t/data=!3m6!1e1!3m4!1surCQBkZmJB46TvOgZp6Hlg!2e0!7i13312!8i6656) seems to give a reasonable match for the buildings in Nogales, with the picture taken the Mexican side of the fence,
Comment: I've been here in person. As of July 2019, the buildings that they are spraying down here still look awful and falling apart.


Explanation for couple not seeing each other before the wedding?

Question: What is the source and reason for a couple not to see one another before the wedding?
Comment: Worth noting that many do not follow this practice, notable among them Rabbis Moshe Feinstein and Joseph Soloveitchik.
Comment: Related: http://judaism.stackexchange.com/q/17760
Comment: What are the source for R. Moshe and R. Soloveitchik?
Comment: @mevaqesh See the article cited below, for one. The better question is why you would assume anyone who knew as much Torah as they did would care about this 'custom'?
Answer: Jewish Action, Summer 2005 edition, has a "What's the truth about..." column by Rabbi Dr. Ari Z. Zivotofsky on not meeting for the week preceding the wedding. His main point is the lack of old sources for this custom, but he does cite several newer sources and the reasons they give. See there for the details, but the reasons and post-facto rationales offered are:

to avoid dam chimud (which he cites strong arguments against as a reason for the custom, so I'll ignore it);
to prevent discussions that will lead to strife (Rabbi E.M.M. Shach; Rabbi Y.S. Elyashiv, who therefore bans communication by telephone also);
to allow the couple time apart for introspection (Rabbi Binyamin Forst);
to heighten excitement (unsourced); and
to prevent premarital relations (unsourced).

And as Rabbi Zivotofsky notes, the last reason should apply only to being alone together, not to simply meeting. Also, see the article for reasons the custom does not exist at all and for rabbis quoted as saying it does not (but also some who defended it).
As always, consult your rabbi for a practical ruling.
Comment: Copy-pasted (with some changes) from [my own answer](http://judaism.stackexchange.com/a/17764) to an older question, which see for additional contributor information.
Comment: Thanks; just can't help myself but to shout out that "those rabbis who say there's no such custom" includes Rabbi Moshe Feinstein (orally, not in writing). Don't get me wrong, it's still the standard custom, and CYLOR.
Comment: @Shalom There is a custom (used to be more popular) to have the bride and groom have a party the night before the wedding https://www.hamichlol.org.il/%D7%97%D7%AA%D7%9F_%D7%9E%D7%90%D7%94%D7%9C


after software update, website not accessible

Question: I just applied a bunch of recommended updates this morning (which seems to have upgraded the kernel to 3.8.0-32.47), and suddenly I cannot connect to the machine on port 80 (http) or 22 (ssh).  Ping still works.  In addition, my syslog and kern.log are filling up with messages like
Oct 22 10:43:27 mu kernel: [ 4041.036862] Inbound IN=eth0 OUT= MAC=6c:62:6d:c8:b0:d3:18:03:73:34:81:b8:08:00 SRC=192.168.212.43 DST=192.168.212.56 LEN=48 TOS=0x00 PREC=0x00 TTL=128 ID=17389 DF PROTO=TCP SPT=59502 DPT=80 WINDOW=8192 RES=0x00 SYN URGP=0

I can connect to the web server locally, so this smells like a firewall issue to me.  I've tried:

doing "reset firewall" from inside GUFW
doing "sudo ufw allow 80" and "sudo ufw allow 22" from the command line

and that is not working.  So I went and looked at "sudo iptables -L", and I'm getting this, but I don't understand the output (I have hidden my employer's domain with "example.com")
Chain INPUT (policy ACCEPT)
target     prot opt source               destination
ACCEPT     tcp  --  ad1.example.com  anywhere             tcpflags:! FIN,SYN,RST,ACK/SYN
ACCEPT     udp  --  ad1.example.com  anywhere
ACCEPT     tcp  --  ad2.example.com  anywhere             tcpflags:! FIN,SYN,RST,ACK/SYN
ACCEPT     udp  --  ad2.example.com  anywhere
ACCEPT     tcp  --  ad3.example.com  anywhere             tcpflags:! FIN,SYN,RST,ACK/SYN
ACCEPT     udp  --  ad3.example.com  anywhere
ACCEPT     all  --  anywhere             anywhere
ACCEPT     icmp --  anywhere             anywhere             limit: avg 10/sec burst 5
DROP       all  --  anywhere             255.255.255.255
DROP       all  --  anywhere             192.168.212.255
DROP       all  --  base-address.mcast.net/8  anywhere
DROP       all  --  anywhere             base-address.mcast.net/8
DROP       all  --  255.255.255.255      anywhere
DROP       all  --  anywhere             0.0.0.0
DROP       all  --  anywhere             anywhere             state INVALID
LSI        all  -f  anywhere             anywhere             limit: avg 10/min burst 5
INBOUND    all  --  anywhere             anywhere
LOG_FILTER  all  --  anywhere             anywhere
LOG        all  --  anywhere             anywhere             LOG level info prefix "Unknown Input"
ACCEPT     tcp  --  anywhere             anywhere             tcp dpt:http
ufw-before-logging-input  all  --  anywhere             anywhere
ufw-before-input  all  --  anywhere             anywhere
ufw-after-input  all  --  anywhere             anywhere
ufw-after-logging-input  all  --  anywhere             anywhere
ufw-reject-input  all  --  anywhere             anywhere
ufw-track-input  all  --  anywhere             anywhere
ACCEPT     tcp  --  anywhere             anywhere             tcp dpt:http

Any idea what I've done wrong?  Or how to fix it?  (I think I'm about to learn a lot about iptables.)
UPDATE
Unsurprisingly, other services on the machine (MySQL) are not responding either.  nmap is telling me that the ports are open, tho
chris@mu:/var/log$ nmap 192.168.212.56

Starting Nmap 6.00 ( http://nmap.org ) at 2013-10-22 10:55 EDT
Nmap scan report for sentry (192.168.212.56)
Host is up (0.00030s latency).
Not shown: 993 closed ports
PORT     STATE SERVICE
22/tcp   open  ssh
80/tcp   open  http
111/tcp  open  rpcbind
139/tcp  open  netbios-ssn
445/tcp  open  microsoft-ds
3306/tcp open  mysql
5666/tcp open  nrpe

Answer: I finally found the command sudo iptables -F which can be used to erase all the firewall rules (it's a dev machine, so I don't really care), and used it. Now everything is accessible.


Should I put a surge protector on a washer or dryer?

Question: I think the question title says it all, but is there a reason that I should or should not put a surge protector on my washer or dryer? Growing up, I don't recall ever seeing them in my parents' house nor at laundromats. But given that this is a several hundred dollar appliance, I'm not sure why these aren't protected by a surge protector.
Answer: If they have the simple electromechanical timers, there's not much that can be damaged by 'reasonable' surges - damage from a lightning strike on your line wouldn't be stopped by a surge protector.
Electronic controls typically do have a surge protector built onto the board, so there's no need for an external one.
Answer: A surge protector can protect from surges from:

Motors or other large devices
Lightning
Incoming power problems

A washer or dryer is actually more likely to cause the first category of problems than to be harmed by it. That still leaves open the question of lightning or other external problems. As a general rule, the protection from a surge protector is primarily for sensitive electronics. Historically, washers and dryers had very simple electromechanical timers. Many newer appliances do have sophisticated electronics - e.g., if your appliance can communicate via WiFi then it has, essentially, a computer inside that could use protection.
However, it is also quite possible that the heavy electrical loads, particularly from an electric dryer, may stress the protective electronics in a surge protector quite a bit. I certainly wouldn't put a washer or dryer on a typical $10 - $25 basic surge protector. (Actually, for an electric dryer that wouldn't even be an option, but for the major appliances where it is possible, I still wouldn't do it.)
I would consider a whole-house surge protector as an option. I am skeptical (most of my experience being with the very logical "load plugged directly into a surge protector") of a whole-house surge protector providing the same level of protection as point-of-use surge protectors, but it should still provide significant protection, and do so for everything in the house, including major appliances.
Comment: Surge protectors *aren't* in series with the protected load.  Tear one down sometime, the hot and neutral rails go right on through like nothin', and there are some MOVs across them.  Therefore, the location of the MOVs isn't critical.
Comment: @Harper I stand corrected on the specifics - you're right, nothing short of a true double-conversion UPS is really in series with the load. But logically to me, having those MOVs on (or "near") the specific circuit should be more effective than sitting off the side of the main panel. On the other hand, a whole house surge protector is a lot cheaper than individual surge protectors all over the house.
Answer: First, if by "surge suppressor you mean power strip or some sort of plug-in dingus, those kinds of cheapies are Right Out.  They are way too  flimsy to carry the power for a major appliance. They also shouldn't be used with air conditioners, microwaves or any heat making appliance.
They do make some commercial-grade plug-ins (e.g. Tripp-Lite), and you can also use units intended for "whole house surge suppression" on a major appliance, simply by locating it near the appliance instead of at the main panel.
You should know that electric dryers have only a big resistor across the 240V poles.  Their motors and electronics sit between one of the poles and neutral.  That is the only place you really need a surge suppressor.
It can be a good idea to put a surge suppressor on appliances which are a source of noise and spikes.  A washer and dryer are definitely two of them.


What is the difference between the Middle Ages and the Dark Ages?

Question: Is there any difference between the "Middle Ages" and the "Dark Ages"?
Comment: You have uncovered the dirty secret of historians, which is that they just make up stuff as they go along and use words and terms that have no certain meaning.
Comment: @TylerDurden :-) Although I wouldn't go so far as to say that feudalism itself never existed; as is _en vogue_ in some modern historiographic circles.
Answer: The "Middle Ages" or medieval period generally refers to the entire time span between classical antiquity and the modern era in Western history. The Middle Ages lasted approximately AD 500 - 1500, from the fall of the (Western) Roman Empire to the fall of the Byzantine Empire and the discovery and colonization of the New World. The Middle Ages are often partitioned into the Early, High, and Late Middle Ages.
The "Dark Ages" in the context of medieval Europe refers to a period roughly synonymous with the Early Middle Ages, AD 500 - 1000, but exact limits have changed reflecting developments in the related historiography. The centuries immediately following the fall of Rome were marked by barbarian invasions, population decline, cultural and economic deterioration, and (to modern historians) a lack of records, hence the "dark" moniker.
This concept originated with the Italian poet Petrarch in the 1330s and referred the whole medieval period, but has since been pared down. However its use is now the subject of controversy among historians, who increasingly consider it misleading and biased by Renaissance era thought.
Comment: Well, if you look at the [Wikipedia article](http://en.wikipedia.org/wiki/Dark_Ages_%28historiography%29), "Dark Ages" in its modern use includes everything up to the 13th century. It is a period with only few written records so that it is "dark" to historians (which is the primary reason for th ename). But that is connected to a general cultural decline of course.
Comment: And here I always thought that the time of the Inquisition was the dark ages...
Comment: I should note the use of "Dark Ages" has evolved and is becoming somewhat controversial. http://en.wikipedia.org/wiki/Dark_Ages_%28historiography%29
Answer: Dr. Susan Snyder, my medieval and ancient history professor, argued that the term "Dark Age" was inappropriate for the early Middle Ages because we still have some records from it and some innovation took place. If we said "Dark Age" in class, we had to be referring to the Greek Dark Age, a period with some actual gaps in the historic record. We lose track of almost all connection between pre- and post-Dark Age writing forms, almost as though people lost the ability to write during that period. (Linear B disappeared entirely, Linear A influences later writing in Cyprus.)
Regarding the Middle Ages, Snyder is an enthusiastic Medievalist, thus her offense to the term "Dark" in reference to it. But most people probably mean the early Middle Ages when they say it.
Answer: Note that "Dark Ages" is a therm with high ideological charge.  Is the manichaean historical vision of the "enlightened" ilustrated from the XVIII century that oposses the Middle Ages (supposedly an age of barbarism) to the movement initiated by the french revolution, as if the Ilustrated Movementet had arisen by spontaneous generation.
Answer: The Middle Ages, is a period in World History that, according to the Gregorian Calendar, began in the immediate aftermath of the Roman Empire's collapse in 476 AD/CE-(more specifically, the collapse of the Western Roman Empire), until 1400 AD /CE, with the emergence of the (Northern) Italian Renaissance. It was a lengthy, prolonged period in world history, which is situated in between the Ancient and Modern eras. The Middle Ages, follows the Ancient era, though precedes the Modern era.
The Middle Ages was divided into 2 phases/stages; the Early Middle Ages, followed by the Late Middle Ages.  The Early Middle Ages begins in 476 AD/CE and ends at around 1050 AD/CE and the Late Middle Ages begins in 1050 AD/CE and ends at 1400 AD/CE. The Early Middle Ages, was also nicknamed, "The Dark Ages", by the Northern Italian Renaissance Poet, Petrarch, while the Late Middle Ages was also nicknamed, "The Age of Scholasticism"-(I do not know which person(s) or originated the nickname, "Age of Scholasticism").
Petrarch coined the phrase, "Dark Ages" in reaction to the dull, primitive, anti-intellectual, pan-religious atmosphere and environment that engulfed and characterized much of the Italian peninsula-(as well as Northern and Western Europe, though to a lesser extent). Petrarch was a Poet who yearned and pined for the nostalgia, sentimentality and glory of Ancient Greece and Rome-(with an emphasis on the latter). For Petrarch, the magnificence and greatness of Ancient Rome was then culturally overshadowed and intellectually obscured by the near universal presence of Christianity and the Roman Catholic Church, in particular. Such a lengthy and primitive overshadowing, was, (when compared with the enlightened character and intellect of Ancient Greece and Rome), a "Dark Age".  The early centuries of Christian Europe-(with an emphasis on the majority of Italy), were, according to Petrarch, the dullest, most anti-intellectual period in recorded history.  Petrarch, from what I know, does not express the same level of contempt for the succeeding Late Middle Ages or "Age of Scholasticism", though was quite vocal in his critiquing of Early Medieval Christianity...a.k.a. "The Dark Ages".
Answer: The "middle ages" are anything between the collapse of Rome (400 AD) to its re-emergence in Rennaissance Italy (1400). Rennaissance means "rebirth".
The "dark ages" are the period in the middle ages which were relatively undocumented. This would definitely include 400 AD to 700 AD, and some extend it to 900 AD.
The "viking period" is 700 AD to 900 AD. Some authors include that period as the dark ages, others do not.
The "medieval period" is either the time between the dark ages and the Renaissance, 700 AD to 1400 AD, or it is the time between the end of the viking predations and the Renaissance (900-1400 AD), depending on the writer.
Comment: Out of the sheer curiosity of someone who is enthusiastic about history but purely on an general culture aspect - why was this answer so heavily downvoted? It does not seem to be substantially different form the others
Comment: I didn't down vote you (I'm coming to this question years later) but it could be because of your dating the beginning of the middle ages  with the collapse of Rome and your dating that at c400. ??? No. Just No. And second using 1400 as the rebirth.??? Again. No.
Comment: @WoJ A lot of the regulars on the forum dislike me, either because I put them to shame, or because I controvert their ideas about history, so they downvote my answers routinely regardless of their quality. I am kind of like the Scaliger of history.stackexchange.com--far ahead of my contemporaries, but hated for it. One guy often copies my answers or pieces of them and then downvotes me if you can believe that.
Comment: well, I would not say that your comments / downvotes on my (amateur) questions were particularly cheerful but I do not see a reason to retaliate by downvoting on what I perceive to be a valid and helpful answer. Must be cultural, I guess .


What was Allah doing before the creation

Question: What was Allah doing before the creation? Is there a Hadith for this? Was God making other words before our own?
Answer: Someone very beautifully answered this question here.
To summarize, Allah is above all constraints like space, time, energy and matter. These cannot be applied to Allah like it does to His creation. Allah was, is and will be. There is no beginning to Him nor an End. So to ask what Allah was doing before / now or after is illogical. If you are asking from the human perspective as to what Allah was doing before He created anything, then the knowledge of the unseen is only with Allah and very little has been given to us (little and good enough to be able to believe in Allah).
Humans have always been curious beings. Islam encourages to question and ponder, but instead of focusing on what Allah was doing, we need to care about what are we doing now in this life to achieve the hereafter. Once faith enters the heart, the clear and the unclear..we accept all of it as it is..as it was made known by Allah to us. The matters of the unseen..if you question too much..there is plenty of room to go astray..we from the outside looking in cannot understand a lot of things now (hell, heaven, angels, death, resurrection)..it is all part of the test. But Allah has given us simple examples to understand some concepts of the unseen so that we can relate to them. Death is like sleep, resurrection is like how the earth turns green after it was once dry, etc..We will see it all for real the moment we die and when later we are resurrected.
Answer: I start with quoting a statement by Musa ibn Jaffar, a descendant of Prophet Muhammad (pbuh) who is considered a scholarly and spiritual authority by Shia Muslims, recorded in a famous theology hadith book, named at-Tawheed as-Saduq:

أنّه قال: إنّ الله تبارك و تعالي كان لم يزلْ بلا زمانٍ و لا مكان و هو
الآنَ كما كان لا يَخلو منه مكانٌ و لا يَشغل به مكان و لا يحُلُّ في مكان
He said: Allah has always been without time or place and He is in
the same state now. No place is devoid of Him, and He
occupies no place and in no place He resides. (source)

This account has several implications:
Allah is essentially beyond time and space.
Therefore, talk of "before" and "after" for Allah is absurd since these two adverbs presume Allah to be subject to time.
It also means God's relation to any point in time is the same as His relation to any other point in time. That is for Allah time doesn't pass. Everything since pre-eternity to eternity are known and present to Him all at once.
Time is indeed an illusion in a deeper sense. What there is is always present! Hence:
What is in store for "future" is already present and known to God. "There is not a thing but with us is its stores" -- Quran.
"Present" is about God presenting to us what has been eternally present to Him but absent to us.
"Past" means something eternally known to God has been just taken away from our sight! So nothing can be actually "lost in history" from an eternal point of view!
It also means, there is a logical possibility that are indefinite number of universes running parallels to ours with their own time and space all simultaneously known to God.
So the question reduces to this: what is God doing in His state of transcendence from time and place?
Answer: Nothing for that would be the station of His essence where there is no change, potential, motion, or expectation. There He only is!
But the traditions suggest that in His eternal state of individuality God desired to be known, hence He created the creation to be known.
Now because we ruled out the possibility of change and motion in God or affection by time, that means Allah eternally wants to be known, so He eternally creates things including time.
So there is no time without creation basically because time is always linked to creation. So we have to say that there is no creation without time but since God constantly creates there will always be time as a byproduct.
So God eternally is, hence He eternally creates. There is no before and after for Him in relation to His creation. Before and after are illusions that affect our limited conditioned existence.
Indeed traditions suggest that for superior beings like some angels time passes so quickly that we can say it's been perhaps like a few seconds since Adam was created depending on how superior the perceiving being is. But for God it would not be even a blinking of an eye! Not even an instant. For God everything only is, no becoming.
Answer: Assalamualaikum
All praise to Allah
The answer is very simple , if you know the theory of big bang before big bang time didn't exist so your question was what was Allah doing before creating us you used the words before and was which is past tense and past tense is connected to time so before big bang time didn't exist
Hope this answers your question.
Answer: Assalamualaikum, may Allah (SWT) let my words from the righteous.
Brother, Allah (SWT) has created time, and space, so he's free from them. Rules are from him, not that he's from the rules. And, to clear out the confusions, after that the creation comes to an end, there's nothing left, Allah (SWT)'s is the only entity left. Allah created us from will to create us, just like we created simulations & games, though not comparing the almighty to us but just for the sake of explanation. Anyways, when there was nothing, the will was of nothing and then was the will to create, so were we created.
Wallahu-Alam.
Answer: What was Allah doing before the creation?
By asking the very question you presume that there are a time span where there are no any matter and change of that matter.
How we know that time passes? By observing (and countining) repeatable events - changes in our world, like raising sun, moon, moving of a watch arrow.
When there is no matter, when there is no things - concept of time become meaningless.
Was God making other words before our own?
About the second question I don't know.
Answer: Allah has many creations, trillions of worlds before this world, and will create many after this world. See Islam Hadith.
See this link: https://myenglishblog.persianblog.ir/x35w0Av9xmHlvB7Ynm8n-Is-there-another-creature-before-Adam-peace-be-upon-him-in-the-ground


getting power to a laptop with a dead battery and broken jack to retrieve info

Question: my dell laptop battery barely holds a charge, the power jack has been damaged to the point that it has just stopped working at all to power the laptop, I know that I need a new computer, but my question is this: can I piggyback to another laptop that is AC powered long enough to power my computer on and get my personal info off of it?
Comment: What do you mean by "piggyback to another laptop"?
Answer: A better solution would be to remove the hard drive from the laptop and plug in to another computer. You should be able to do this either directly (ie. inside the case) or purchase a USB external drive case (or dock) to plug the drive in. This should allow you to read all the data contained on the drive.
Answer: The three possible ways to power a laptop are:

AC adapter to laptop power socket
A fully charged battery
Docking station

No, it's not feasible to piggyback another laptop because the power requirements of each laptop match their power source. So if you were able to jury rig some hypothetical connection, the result would require more power wattage than the working laptop could safely supply.
Drac's suggestion is the most reliable. Another more time consuming way, is to power a charged battery (of the same battery model) to use while copying files. This way might require multiple charge/use/replace cycles.
Using a compatible power adapter would be better, but a common problem that is sometimes reported as a bad power adapter, is really a broken power socket in the laptop. (Where the power adapter plugs into the laptop). This is a point of much physical wear and the solder points on the motherboard can become intermittent.
Good luck.


Why is fluorine the most electronegative atom?

Question: It seems related to the atomic size but hydrogen has a smaller atomic size than fluorine. Why is fluorine the most electronegative atom?
Comment: It's interesting to note that if you define electronegativity as the "strength" with which the valence electrons are attracted to the nucleus (for a certain definition of strength), then the noble gasses also have very high electronegativities. In particular, *neon* becomes the element with the highest electronegativity, since it has the same number of shells as fluorine while having an extra proton! It's kind of odd that neon should be more electronegative than fluorine, because we often associate extreme electronegativity with high reactivity, but it makes a lot of sense.
Comment: One reason is hydrogen has one *shell* and one proton, as compared to fluorine with 2 *shells* and 9 protons.
Comment: Electronegativity is the tendency of an atom to attract a shared pair of electrons (not the valence electrons ) towards itself
Answer: Fluorine is the most electronegative element because the definition of electronegativity makes it so. The electronengativity scales are defined based on experimentally determined properties of the elements. Fluorine has appropriate values for all of the common scales to ensure it has the highest electronegativity.
The Pauling scale, which is the first electronegativity scale proposed, relates the difference in electronegativity $\Delta\chi_{A,B}$ between two atoms $(A\ \& \ B)$ is related to the bond dissociation energies $(E_d)$ of the diatomic species $AA,\ BB,\ \& \ AB$:
$$\Delta \chi_{A,B}=|\chi_A - \chi_B| = (\text{eV})^{-1/2}\sqrt{E_d(AB)-[E_d(AA)+E_d(BB)]/2}$$
Since only differences can be determined, one element (hydrogen) was chosen to have an arbitrarily set value. On the Pauling Scale Hydrogen has a value of $2.20$. Since the $\ce{H-F}$ bond is much stronger $(E_d=586\ \text{kJ/mol})$ than the average of the  $\ce{H-H}$ bond $(E_d=436\ \text{kJ/mol})$ and the $\ce{F-F}$ bond $(E_d=157\ \text{kJ/mol})$. Data from here. In general, since fluorine forms stronger bonds to all other elements than it does to itself, fluorine will have the highest electronegativity.
The Mulliken scale relates electronegativity to two other properties: electron affinity $E_{ea}$ and ionization energy $E_i$:
$$\chi_A = \frac{E_{ea}(A)+E_i(A)}{2}$$
Fluorine has a high electron affinity (easy to gain an electron) and a high ionization energy (hard to lose an electron).
There are other scales, including the Allred-Rochow scale, based on effective nuclear charge and covalent radius, the Sanderson scale based on atomic size, and the Allen scale based on the average energy of the valence electrons.
Answer: Fluorine has a higher electronegatevity if compared with other than elements in the periodic tables due to:

atomic size is decreased.
nuclear charge increase due to gain two electrons.
attraction force between nucleus and adding electron increase so electronegatevity increase too.
Answer: It seems that (although I am no chemist) that the valence electrons in the 2nd shell have a higher priority than any other shell, if the 2nd shell is the valence shell. The closer it gets to being full, we know that electronegativity increases. With Fluorine only needing one more electron to be complete. The big battle for second place is interesting. Oxygen only needs 2 electrons in its valence shell. Chlorine only needs one,  but in its 3rd shell. But we know Oxygen wins out by a slim margin.


Finding passenger lists for arrivals to Baltimore in 1900s-1910's?

Question: I know someone who arrived to Baltimore directly (without going through Ellis Island) in the early 1900s.
Are there any listings of passengers on these ships?
Answer: Ancestry.com indexes Baltimore arrivals along with many other ports. You will need a US subscription to search these databases.
Steve Morse provides a form for searching Baltimore arrivals (if you have access to Ancestry). You may also browse images on the FamilySearch site for free, but it helps if you know the arrival date.
UPDATE ON METHODOLOGY
I've used two different techniques to find records on a specific topic such as Baltimore arrival records. One approach is to find some specific matching individual, and then browse the collection from there. For example, search for any record with an arrival in Baltimore. This works well on Ancestry, which has a rich query form.
The second approach is use the catalog to find a desired collection, and then to search that collection. To find collections one can use the catalog on FamilySearch or the card catalog on Ancestry.com. Catalogs can be searched by keyword (e.g., immigration) or location (e.g., Baltimore) and then the desired collection can be identified by browsing. Once a collection is found, it generally comes with a search form that restricts search to that collection, or (in the case of FamilySearch) a form for ordering documents from it on microfilm.
Answer: For finding lists of arrivals to the USA, the process has been greatly simplified by the work of two individuals.  As noted in this answer, Stephen P. Morse's One-Step Web Pages are search portals that are usually more sophisticated and straightforward than the forms provided at the provider's own websites. Joe Beine's guides, published on his website German Roots, give overviews of what records exist and how to find them.  For the period you want, try his Baltimore Passenger Lists Quick Guide 1820-1950s.
Having online access via Ancestry is convenient, but if you don't have a subscription, and can't access Ancestry at a library, you have other options.  You can visit a National Archives facility, where you can view the microfilms for free, and you can search or browse the records at FamilySearch.org (see links below), or browse (or download) copies of the microfilms via the Internet Archive.
When you are searching for records in their original form, sometimes the records themselves are on one set of microfilms, with an index (which would have been created separately) on a different set of microfilms.  Searching for a particular arrival on microfilm is a two-step process which begins with using the index to find out which specific roll of microfilm you want from a large group, then working with that particular roll of film.
Using the German Roots website
Beine's guide simplifies the process by providing a cross-reference of the information about the original records from NARA (the National Archives and Records Administration) and the FHL (the Family History Library). Beine also gives a direct link to the collection on Ancestry, so you don't have to go through the card catalog. For 1900-1910, these are the appropriate sub-pages on German Roots:

Baltimore Passenger Lists Index 1897-July 1952
Baltimore Passenger Lists 1891-1948

Information from NARA
Both record sets are part of Record Group 85, Records of the Immigration and Naturalization Service, 1891-1957.  NARA's descriptive pamphlets on the records are here:

Index (Soundex) to Passenger Lists of Vessels Arriving at Baltimore, MD, 1897-1952. T520. 43 rolls 16mm. PDF
Passenger Lists of Vessels Arriving at Baltimore, MD, 1891--1909. T844. 150 rolls. (Rolls 78-150 are new listings.) The manifests were filmed as they appeared in bound volumes. Date spans may overlap. PDF

NARA's online catalog gives information about each microfilm publication, including a list of NARA facilities where the microfilm can be viewed for free.  The relevant entries are:

Publication T520, Index (Soundex) to Passenger Lists of Vessels Arriving at Baltimore, Maryland, 1897-July 1952.
Publication T844, Passenger Lists of Vessels Arriving at Baltimore, Maryland, December 1891-June 1948.

Using FamilySearch
On FamilySearch.org, the links on the Research Wiki which describe the records are:

Maryland, Baltimore Passenger Lists Index (FamilySearch Historical Records)
Maryland, Baltimore Passenger Lists (FamilySearch Historical Records)

The links to the collections themselves in the FamilySearch Catalog are:

Maryland, Baltimore Passenger Lists Index, 1820-1897
Maryland, Baltimore Passenger Lists, 1820-1948

From here you have the option of searching by name, or choosing the Browse link to step through all the images, as if you were viewing the microfilm.  The online images have waypoints which correspond to the beginning of each roll of microfilm.
Clicking Learn More on any of the catalog pages is supposed to take you back to the relevant articles on the Research Wiki, but this feature is not working reliably (at least when using Firefox).  The links to those articles are in this answer, and articles can be searched for by going directly to the Wiki and putting your subject into the search box.
Using the Internet Archive
The Internet Archive has (at the moment I write this) 636 items in the category Passenger Lists of Vessels for Baltimore and Philadelphia -- the entry page is here: Welcome to Passenger Lists of Vessels for Baltimore and Philadelphia.  Clicking through to the full list of results will give you a list of titles -- many of them are the individual rolls of microfilm, with the film roll number in the title of the catalog entry.  Once you know which roll you need, you can read the film online, or download a PDF for offline reading.
Using Ancestry
An online version of the records is held at Ancestry.com, as the database Baltimore, Passenger Lists, 1820-1964 (requires subscription).  On the main search page for any collection, more information about the collection can be found by scrolling down to read the sections Source Information and the About [database title].
One-Step Web-Pages
Here are the links on Morse's page which relate to Baltimore arrivals. If you haven't used his search forms for passenger lists before, read his FAQ (there's a button near the top of the Baltimore Passengers page).

$ Baltimore Passengers (1820-1948): Searching the Baltimore Passenger Lists in One Step links to Morse's page for searching for passengers at Ancestry, pre-loaded with the information for Baltimore.
Accessing Baltimore Manifests in One Step (from 1820 to 1948) gives the users controls for browsing the microfilm at Ancestry.com directly.
Ship Lists: Searching for Ships in the Baltimore Microfilms in One Step is used in combination with the direct manifest link to locate the microfilms belonging to a particular ship.


Opinion on a symbol

Question: I would like to create a new symbol for WLOG (= without loss of generality), since there doesn't seem to be any around. I have some variants and I'd like a 'vote of confidence' on either from the community. Would you close such question? (Before you answer, please take a look at: Is there another symbol that is slightly different from \forall (likewise for \exists)?)
Comment: My opinion is that the symbol should be "without loss of generality", completely spelled out. Symbols such as "for all" are *not* abbreviations.
Answer: Since the title is now "Opinion on a symbol", I'd like to give mine.
Symbols in mathematics are not abbreviations. Using \therefore or \because in a mathematical text is wrong, unless they are used in a formal proof with a well defined meaning. But plain mathematical proofs are usually not formal proof: they should be understandable by the reader and symbols should denote mathematical entities, not linguistic ones.
So I never use \forall or \exists in a proof; sometimes I indulge in a \implies or \iff in a set of displayed formulas, but very sparingly and only when they convey a mathematical, rather than linguistic, meaning.
One of my first duties when converting to LaTeX some papers written by a friend and colleague was

find "w.l.o.g." and replace it with "without loss of generality"

She knew about this and accepted it, because otherwise she'd have had to learn LaTeX. :)
Do a favor to your readers and use words rather than shorthand.
Finally, in a formal proof there's no way of saying without loss of generality: a formal proof must check all possibilities. So even there we can't find a use of a symbol.
Comment: I have to slightly disagree with you. I think that text abbreviations like "w.l.o.g." and "iff" can be used in a mathematical text and don't make it inconvenient. They are well understandable and really common. But I suppose that this may differ from comunity to comunity.
Comment: @tohecz Actually no: *if* is improper (not to say wrong). One should say "X stands for A, B and C" or "X is defined to mean A, B and C" or the like. So the use of a different word is justified.
Comment: Thanks @egreg - I already got strongly discouraged from doing that at the math.SX site. I saw it recurring in an algorithm description with several agents and the idea is that any agent can make the first move, without changing the overall outcome. I agree, WLOG is rather casual... and I don't really feel comfortable using it.
Comment: @tohecz "Iff" was invented, IIRC, by Halmos for stating *definitions*, where the use of "if and only if" would be too heavy. Unfortunately it has since been abused where "if and only if" would be necessary *and* clearer.
Comment: Sigh, for definition, simple _if_ ought to be used imo.
Comment: Incidentally, this idea that symbols are not abbreviations is somewhat field-dependent. It applies in pure mathematics, of course, but in other fields (e.g. physics) such symbols are often used to indicate something similar, but not identical, to their pure mathematical meaning. In any case the issue of whether or when a particular symbol is appropriate is kind of tangential to the main point of the question.
Answer: I think it's better to ask a specific question like the one you linked to. Something like, "How would I create the following symbol that would represent 'without loss of generality'?":

And then include an image like above, together with a sample usage and possible dimensions. A generic question about the symbol might not lead to it being helpful for future users, since the final choice of the "most helpful" answer could be very subjective. The StackExchange network - in most cases - doesn't works much better in a definitive Q&A style than asking questions that seek generic advice on topics. Perhaps also refer to Don't ask us about best practices, glom of nit, or Mrs Cake.
Answer: You could define (this assumes that you're using the amsmath package and its \text command):
\newcommand\wlog{\text{\textsc{wlog}}}

Obviously this isn't a symbol (at least not in any sense of the word I'm familiar with), but it would have the advantage of being easily parseable.

Opinion on a symbol


How to reset PC from FPGA (PCI board)

Question: How to reset PC(motherboard) from FPGA on PCI board (without BIOS manipulations)
Subj.
Answer: You are probably not going to find a scheme that works with all motherboards, if you even find any.  Intel and AMD do this differently too.  If you're lucky to find a motherboard where the PCI reset signals are open drain you might be able to drive it and cause some bad stuff to happen and cause a reset.  If the motherboard drives those signals then you are out of luck (you could damage the driver).
You could also try to crash the system that would cause a reset, although that's getting harder nowadays with PCIe.  Operating systems seem to be able to survive bad PCI devices due to better error handling and timeouts, etc.
Is this to implement some sort of watchdog?  Probably easier to do this using IPMI and some sort of heart beat.

How to reset PC from FPGA (PCI board)


Why do salt flakes start building up in a bottle of mineral water after it was first opened?

Question: I noticed that mineral water with high mineralization doesn't have salt flakes floating on the bottom of the bottle on store shelves, but when I open one, after some time they start accumulating. Does it have something to do with decarbonation?
Answer: http://en.wikipedia.org/wiki/Solubility
Salt solubility decreases with decreasing pressure.
Answer: It could be gases (oxygen or carbon dioxide) from the air, competing as solvents with the salts.  I know the reverse effect: adding salt tends to drive out gases.
I'm not sure whether this gas explanation is more significant than the pressure explanation.
But, you could do an experiment: briefly let the pressure out and then retighten the cap without letting fresh air enter, and see if the crystals still form.
Comment: They do, since the pressure falls.


Sink the Submarine

Question: There is submarine somewhere on the number line. Initially (say, at midnight), it is at position $A$, and is moving $B$ units per hour to the right, where $A$ and $B$ are both integers; both $A$ and $B$ are unknown to you. You have an unlimited number of torpedoes. Starting at midnight, and at the top of every hour, you can fire a torpedo somewhere on the number line, sinking the sub if it is there. What strategy can you use to guarantee you eventually sink the sub?
Comment: @Allan [A submarine problem post on math stack exchange](http://math.stackexchange.com/questions/642957/a-problem-related-to-the-submarine-puzzle), which contains a solution where $A,B$ real, but the torpedo destroys an interval as opposed to a point.
Comment: @BigAgnes Yes you are correct. After coming back to this problem after some time, I've come up with an actual algorithm that can enumerate positive and negative rational numbers; however it takes a while for large numbers when taken to their absolute value. I'll add it into my answer below.
Comment: Can A and B be either positive or negative?
Comment: @JoeZ. Yes they can, and a negative value of B means the sub is moving left
Comment: @MikeEarnest, this question also exists as duplicate on Math.SE. You can also sink the sub if its speed and starting position are not integers, if the torpedo has a (fixed) target range.
Comment: @Aravind I don't think it's possible if it's speed and starting position are not integers. Also, could you add a link to the duplicate on Math SE?
Comment: @MikeEarnest I see now, thank you. I missed the point about the torpedo destroying an interval.
Comment: @Allan In theory, it should be possible even if $A$ and $B$ are rational, yes?
Answer: We are given the following information
P(0) = A
V(t) = B
P(t) = A + Bt
Since we know our current time, we can cycle through all the possibilities of P(t) by cycling through all the combinations of A and B at the current time and we will eventually land on the correct combination of A and B.
NOTE: There are actually an infinite amount of combinations because the set of integers are unbounded. But since we know that A and B are finite numbers, this algorithm will eventually find the answer.
Here is a picture which shows a method for computing all the possible combinations. I'm sure there's probably an actual function out there that is able to compute the unique pair of values given an integer.

NOTE: The image depicted here is not the exact same as the function used in the solution below (it's simply been rotated). To view the actual geometry of the function used, visit the link.
Note that we have to increment t by 1 each time because we are firing the torpedo with respect to the current time (1 hour intervals).
I'll try to come up with a better explanation and maybe an animation if I have the time.
EDIT: (Sketchy) Algorithm for enumerating 2 rational numbers:
Since enumerating 2 rational numbers would essentially require enumerating 4 integers (i.e. integers a,b,c,d correspond to rational numbers a/b and c/d), this is what this algorithm essentially does.
First add 0 (special case that only needs to be considered once)
Then find all permutations where 1 is the biggest

1 1 1 1

Then find all permutations where 2 is the biggest

2 1 1 1
  ...
  2 2 2 2

and so on...
Now for each line we need to enumerate all possible permutations of +/- (2^4 permutations).
Comment: The visual really drives the point home, well done!
Comment: Thanks! I also just figured out that the functions used for A and B in my solution must be wrong since they will always yield a positive integer (thus their combinations only cover the upper-left corner of the grid). I'm sure that somewhere out there, someone else has already figured out a function for it. I guess manually drawing out the graph and crossing out the values as they get used would work as well. I was hoping for a function that would be computationally simple :/
Answer: There are countably many possible submarine paths, since each is given by an integer start point and speed. Enumerate all of them. On hour $h$, sink submarine $h$ on the list by bombing its location at that time. Any possible submarine will eventually be be sunk.
I could give an explicit strategy, but I find that brute-force constructions like this better explain why a solution must exist than clever explicit ones.
The same construction extends to submarines that move with as integer quadratics, or integer polynomials, or move with rational speeds, or move on lattice points on the plane, etc.
Comment: Dam you came up with the idea before I did :)
Answer: It's one ugly formula, but at least it's explicit! You can fire your $n^{th}$ torpedo at location
$$P(n) = (-1)^n\left[A(n) + B(n)\cdot n\right]$$
where
$$A(n) = \left\lfloor\frac{ \left\lceil\frac{\sqrt{8n+1}-1}{2}\right\rceil\left(\left\lceil\frac{\sqrt{8n+1}-1}{2}\right\rceil+1\right) - 2(n-1)       }{4}    \right\rfloor$$
$$B(n) = \left\lfloor\frac{2(n-1) + 2\left\lceil\frac{\sqrt{8n+1}-1}{2}\right\rceil -\left\lceil\frac{\sqrt{8n+1}-1}{2}\right\rceil\left(\left\lceil\frac{\sqrt{8n+1}-1}{2}\right\rceil+1\right)}{4}    \right\rfloor$$
For instance, this table gives $P(n)$ for the first ten shots (and a few more).
<code>Torpedo (n)     Location P(n)     Corresponding (A,B)
1               0                 (0,0)
2               1                 (1,0)
3               3                 (0,1)
4               -1                (-1,0)
5               6                 (1,1)
6               -6                (0,-1)
7               2                 (2,0)
8               7                 (-1,1)
9               -8                (1,-1)
10              20                (0,2)
...             ...               ...
35              -104              (1,-3)
100             -397              (3,-4)
1000            5018              (18,5)
12345           -604934           (-29,-49)
</code>
Derivation
Where to fire
Clearly, if we know $A$ and $B$, then at time $n$ we fire our torpedo at position
$$P(n) = A + Bn$$
Mapping $\mathbb N$ to $\mathbb Z$
This function provides a bijective map from $\mathbb N$ to $\mathbb Z$.
$$H(k) = (-1)^k\lfloor k/2 \rfloor$$
For instance, we have $$H(1) = 0 \quad\quad H(2) = 1  \quad\quad H(3) = -1 \quad\quad H(4) = 2 \quad\quad H(5) = -2 \quad\quad etc$$
Enumerating $A$ and $B$
Consider a table, where the $i^{th}$ row and $j^{th}$ column holds the pair $(H(i), H(j))$. Every pair $(A, B)$ lies somewhere on this table, so we need only find the $i$ and $j$ which correspond to it. We will map $\mathbb N$ to $(A,B)$ pairs by moving along the diagonals. For instance, the first $5$ $(A, B)$ pairs are
$$(0,0) \quad\quad (1,0) \quad\quad (0,1) \quad\quad (-1, 0) \quad\quad (1,1)$$
Formally, this involves mapping $n$ to a pair $(A, B) = (H(I(n)), \ H(J(n)))$.
Finding $I(n)$ and $J(n)$
Let $T(k)$ be a function which returns the $n$ corresponding to the first row and $k^{th}$ column of our table. This function can be written recursively as
$$T(k) = T(k-1) + k \quad\quad \text{with }\ T(0) = 0$$
This recursion relation has the solution
$$T(k) = \frac{k}{2}\left(k+1\right)$$
whose inverse is
$$T^{-1}(k) = \frac{\sqrt{8k + 1} -1}{2}$$
Since the $k^{th}$ diagonal contains $k$ pairs, the $k^{th}$ diagonal of the table contains the $u^{th}$ through $v^{th}$ pairs where $u = T(k) - k + 1$ and $v = T(k)$. Moreover $\lceil T^{-1}(n)\rceil$ gives the length of the diagonal which contains the $n^{th}$ pair. Therefore in terms of $n$
$$v = T\left(\lceil T^{-1}(n)\rceil\right)$$
$$u = v - \lceil T^{-1}(n)\rceil + 1$$
Now just by counting rows (or columns) away from $u$ (or $v$) we obtain $I(n)$ and $J(n)$.
$$I(n) = v - n + 1$$
$$J(n) = u - n$$
Putting all the pieces together gives the equation for $P(n)$.
Comment: I'm a little confused with what's going on with T(x). Could you also add in some brackets into T(x) and it's inverse function (just to clear up what's getting multiplied)?
Comment: Sure, T is a little confusing. It makes more sense when you think about the recursive version of the formula. The explicit version is just a solution to that recursion (Which is far less intuitive).

Sink the Submarine


Proper Time with a non-zero Initial Velocity

Question: I have a question regarding how to find the proper time for a body with an initial velocity to slow down to 0. For example, the equation I have been working with looks like:
$$\int^\tau_0 d\tau=\int^t_0\frac{1}{\sqrt{1+\frac{(\alpha t+v)^2}{c^2}}}dt$$
Where $t$ is equal to the coordinate time for the rocket to slow down, $\alpha$ is the proper acceleration and $v$ is the intial velocity. Integrating gives:
$$\tau=\frac{c}{a}arcsinh(\frac{\alpha t+v}{c})$$
I believe, however, that this is incorrect as the values it returns are far too small to be a legitimate value for proper time. I do believe that this is on the right track to deriving the correct value.
Answer: Your formula is almost correct. If $\alpha$ is the (constant) proper acceleration and $v_0$ the initial coordinate velocity, then
$$
v(t) = \frac{\alpha t+w_0}{\sqrt{1+(\alpha t+w_0)^2/c^2}},
$$
with
$$
w_0 = \frac{v_0}{\sqrt{1-v_0^2/c^2}}.
$$
See this post for the derivation. The proper time is then found by integrating
$$
\text{d}\tau = \sqrt{1 - v^2/c^2}\text{d}t = \frac{\text{d}t}{\sqrt{1+(\alpha t+w_0)^2/c^2}},
$$
thus
$$
\tau = \int_0^t\frac{\text{d}t'}{\sqrt{1+(\alpha t'+w_0)^2/c^2}}.
$$
If we use the substitution $y = (\alpha t'+w_0)/c$, we get
$$
\tau = \frac{c}{\alpha}\int_{w_0^{\phantom{1}}/c}^{(\alpha t+w^{\phantom{1}}_0)/c}\frac{\text{d}y}{\sqrt{1+y^2}},
$$
and that's where the problem was: the lower bound is not zero. The result is
$$
\begin{align}
\tau &= \frac{c}{\alpha}\left[\text{arcsinh}\left(\frac{\alpha t+w_0}{c}\right) - \text{arcsinh}\left(\frac{w_0}{c}\right)\right]\\[0.4cm]
&= \frac{c}{\alpha}\left[\ln\left(\frac{(\alpha t+w_0)/c + \sqrt{1+(\alpha t+w_0)^2/c^2}}{w_0/c + \sqrt{1+ w_0^2/c^2}}\right)\right].
\end{align}
$$
Comment: Hmmmm, alright I believe I understand it for the most part. Is there any distinction between the $\tau$ and $t'$ you use in the integral?
Comment: @HattarWu $t'$ is just a name for the coordinate time in the integration, to avoid confusion with $t$ in the upper bound.
Comment: Ah, I see. For the coordinate time used in the final equation, is that equivalent to the coordinate time used in the first equation for $v(t)$? Wouldn't that imply that, because the rocket is slowing to a stop, $t=\frac{-\omega_0}{\alpha}$ where $\alpha = -(2*9.81)$? That feels like it does not take into account relativistic effects.
Comment: @HattarWu Yes. If you insert that value, the expression simplifies a lot.
Answer: Yeah, something there is incorrect. I imagine $t\rightarrow\infty$, and see that $\alpha t+v$ will $\rightarrow\infty$ too.  But, unless you're using some weird definitions, always $v<c$.
Given proper acceleration $\alpha$, the way to make use of it is from its definition, and of the four-velocity, proper time, and Lorentz transforms.  The rocket's velocity at some time $t+\delta t$ should be $\alpha\delta\tau$ as seen in the rocket's own frame at time $t$. Its own change in velocity over a short duration of time as it knows it.
Write out the relativistic four-velocity, think about the relation between $\delta\tau$ and $\delta t$, and of course we're thinking of small time intervals, $\delta t \rightarrow 0$ - it just seems natural to write a differential equation.  Have you done that?  Where did that integral of yours come from?
There's the concept of "rapidity" which seems to come up more in high energy physics than in other areas, you might find enlightening to read about.
Comment: The integral comes from the definition of the Lorentz factor: $$\gamma = \frac{dt}{d\tau}$$
$$d\tau = \frac{dt}{\gamma}$$
and then substituting the version of the Loretnz factor with its definition in terms of $\alpha$ and integrating. I'm sorry, but I don't quite understand exactly what you are saying....what exactly is the problem with the equations? I know there is a problem, but I can't figure out where it is!