Usefulness of launching probes to verify the existence of a hypothesized planet

Question: In a parallel universe, humans have a 95% confidence that Planet 9 exists with the parameters given on Wikipedia. However, they have not directly observed it and plan on launching many space probes to different regions of the Solar System. Each probe costs $\sim \$30 \text{ million USD}$ and requires at least $18 \text{ km}\cdot \text{s}^{-1}$ of $\Delta v$ to reach the hypothesized Planet 9. As they do not know where exactly Planet 9 is in its orbit, they plan on launching them at $1^\circ$ intervals. Is this feasible, and at what timescales and technologies can achieve this?
Comment: So this is about a planet that astronomers can't detect but suspect exists in out solar system? There are lots of large asteroids/dwarf planet bodies in the Khyber belt. Why would they care about this enough to spend money trying to find another Pluto-like world? We need to understand why it's important to put a value on finding it.  If they know a proposed orbit, they would run a single probe to the proposed orbit, flying along it in the opposite direction of suspected planetary motion until they came across it or didn't. There would be no hurry unless there was something special about it.
Comment: @DWKraus Remember that [judging the back story](https://worldbuilding.meta.stackexchange.com/q/8008/40609) is really only relevant when it directly impacts the possible answers of the question. In other words, a question shouldn't be judged on the basis of why an OP wants an answer unless it can't be answered without knowing why (I don't believe this one does...).
Comment: @DWKraus I don't want to argue about it, but no... it doesn't. It's the OP's world and the question is only referring (as I read it) to technical feasibility. Our site sometimes has trouble looking past the back story to enjoy answering the question.
Comment: Also, what tech level are we talking about? Today tech? Today, they wouldn't spend much at all because they wouldn't care. In the future, the probes would cost less and do more. Plus I don't know if a probe would be the best tool for this. Why can't they detect the planet with conventional tools from Earth?
Comment: @NuclearHoagie The planet's existence has been postulated based on the orbits of other Kuiper Belt objects it may have perturbed in the past, meaning that its orbital parameters can be estimated but its present location is not known to a high degree of accuracy.
Comment: A really cool read is "[Lighting the Path to Find Planet Nine](https://earthsky.org/space/lighting-path-to-find-planet-nine/)." The answer to your question is very likely "yes, it's feasible," but considering the amazing things astronomers can do without probes, it might not be practical - but that wouldn't be any fun.
Comment: @JBH They want to know the feasibility, time scales and tech needed. All these things call for judgements about why people are doing it. Short answer is: they would never spend the money to do something they could do slower and more cheaply, so it wouldn't work. They would do something slower and cheaper. OR we could launch in six months and be there in a year if it were the source of an alien invasion or it's made of rare-earth metals and the first probe to prove it claims it.
Comment: $30 million price tag is for the probe only, or probe and launch vehicle total?
Comment: "According to University of Michigan professor Fred Adams, **within the next 10 to 15 years, Planet Nine will either be observable or enough data will have been gathered** to rule out its existence.[137][138]" #137: "Until the next generation of super-space **telescopes** makes it into orbit, Planet Nine is probably going to remain a theory, but you can help. The Backyard Worlds: **Planet 9 project is calling on citizen scientists to join the search** for 'cold worlds' orbiting the sun by browsing images from NASA's WISE satellite, which is currently surveying the entire sky in infrared."
Comment: ... So, we don't need any probes, just a few new telescopes, and or, finish sifting through the data we already have from WISE and their ilk which already ruled out Nemesis. Number nine is next.
Comment: @Mazura There's already evidence that it's observational bias, so the chance of Planet 9 actually existing is smaller than thought before.
Answer: It sounds like a poorly conceived waste of money.
Launching many space probes with 1 degree separation means launching 360 probes. If we take for good the lowest value of the range for the orbital period, 10000 years, we get that each probe would have to cover a span covering about 28 years of the orbit of the planet. Basically the luckiest probe would reach the orbit of the planet somewhere between right on time to meet it or 14 years too late/early.
Moreover, not knowing where the planet is, one would need to account for considerable extra fuel to allow orbital maneuvers which cannot be designed nor drafted before launch, further complicating the mission. And mind that this assumes we know for sure what the orbital plane of the planet is.
With 10.8 billion dollars (360 times 30 million) it would make more sense to build a dedicated orbital telescope to scan the sky looking for the planet in both visible and IR spectrum.
Comment: 360? ONLY if you assume the planet is exactly on the ecliptic. Try 41253 probes, if you want to cover *every* 1 degree angle out there.
Comment: @PcMan that's a whole sphere, do support such efforts, but it is overkill. There are some ideas where the orbt should be, it is not completely shoot in the dark
Answer: There are three main issues: Cost, low probability of success, and better options.
Current observations have attempted to constrain the current location of Planet Nine to within a region of sky roughly 20-40$^{\circ}$ across (Fienga et al. 2016, Holman et al. 2016). That's not a small portion of the sky (although at least it's not a completely blind search. . .). If you were to send probes out spaced 1 square degree apart, you would need hundreds of them (say 900, assuming a lattice 30$^{\circ}$ on a side), leading to a cost of ~\$30 billion just to construct the fleet, let alone operate it - after all, you'll have to receive the data being transmitted by the probes.
The Voyager probes have travelled 120-150 AU in the 44 years of their existence. Planet Nine is expected to be near its aphelion point a bit over 600 AU from the Sun, meaning it would take around a century and a half to reach its target - but during that time, the planet's location would have changed by at least a few degrees, given its estimated orbital period. This adds to the issue that we'd need to know its orbital parameters well enough to predict its location, which could be difficult. You'd still have a similarly-sized region of the sky to search, just shifted a bit.
There's also no guarantee that these probes would survive the journey in working shape. Probes in space suffer from space weathering, and the instruments would have to be well-shielded. Some of them will break down along the way, which could include transmitters needed to communicate with Earth and send back data. This also assumes that the launch and voyages go smoothly. If you launch hundreds of presumably decent-sized satellites, you'll need scores of rocket launches, and some of those rockets will explode. That's not a good look for a program in need of massive government funding.
I agree with L.Dutch that it would be more effective to focus on building better telescopes closer to home. The cost would be much lower; the Subaru Telescope, currently being used to search for Planet Nine, cost about \$400 million dollars. Even the most expensive ground-based observatory, ALMA (a radio telescope, and only listed here for cost estimates) only cost \$1.4 billion dollars - about 10% of what I'd bet these probes would cost. Moreover, non-specialized instruments like Subaru and ALMA can be used for many other projects which are almost certainly guaranteed to yield results. Astronomers (and the public!) will be much more eager to support something that can effectively guarantee a high return on investment.
Another possibility is to launch a smaller fleet of probes but attempt to perform a deep sky search closer to the location, rather than just stumbling upon the planet, which seems to be the current plan. In other words, you'd be replicating current observing procedures from a closer distance. Unfortunately, you'd have to problem of launching a series of telescopes into space that would be similar in size to Subaru, which is 8 meters across, and that's unlikely. It's honestly really difficult to put large telescopes in space. JWST costs about \$10 billion, and that's just one telescope, a bit smaller in size! So this probably isn't a great alternative.
Answer: Let's run with this idea!

Nothing makes you feel like a man more than pulling the trigger in an Abrams M-1 tank! (Attribution: a friend of mine about 20 years ago)

Let's assume that the scientists on Antarctica have broken away from their respective countries and declared themselves to be independent! To prove that, the newly-established Oligarchy of Antarctica has created a unified space agency that wants to prove itself to the world. Like the space race between the USSR and the U.S. back in the 50s and 60s, it's looking for some serious bragging rights. Yes, astronomers can detect some amazing things without ever leaving the orbit of Earth...
But nothing says "show us respect!" like launching a rocket!
However, there's one problem that makes launching 360 rockets almost irrelevant.

"Space," [The Hitchhiker's Guide to the Galaxy] says, "is big. Really big. You just won't believe how vastly, hugely, mindbogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space..." (Douglas Adams)

But we do have a benefit! Current theories believe Planet 9 is whomping huge! According to this article...

Nobody has yet found Planet Nine. If it exists, it's large and very distant. But there are reasons to believe it does exist (and reasons to believe it doesn't), and so for astronomers – and citizen scientists – the hunt has been ongoing for several years. If it exists, Planet Nine would be a so-called super-Earth, a planet with a mass higher than Earth's, but substantially below those of the solar system's ice giants, Uranus and Neptune. According to Yale professor of astronomy Gregory Laughlin, senior author of the new study, Planet Nine would have five to 10 times the mass of Earth and be located hundreds of times farther from the sun than Earth, and 14 to 27 times as distant from the sun as Neptune.

So, 360 probes heading out 14-27 times the distance of Neptune... to understand the distance between the probes, go re-read that quote from Douglas Adams.
But, what if we changed things up a bit? RADAR that far out in space wouldn't have as many issues as RADAR on Earth, where a pesky atmosphere and rudely inconvenient solar winds get in the way of anything really useful. Out in space you'd have less beam dispersion and much less trouble expecting a return.
Now, we're looking for suspension of disbelief, not exactly real-life. In real life what you're proposing is impractical to a degree that makes building the pyramids in Egypt look like playing with Legos. So, let's simplify RADAR a bit and assume that what we really care about is the maximum velocity of the probe to allow a bounced beam to be recovered at a distance of, say one billion km. That wouldn't cover all the space we'd like, but it would cover the majority of it. So, let's roll!
Assumption: we have the ability to transmit a signal that won't degrade so much in the time we're talking about such that we can trust that we'll get a good RADAR reading. Considering we're still getting signals from the Voyager probes, this might be implausible, but it doesn't seem impossible.
So, the speed of light (aka RADAR) is 300,000 km/s and we want to detect something 1,000,000,000 km away. That's a round trip of 6,667 seconds or just over 1.85 hours. Planet 9 could be a whole lot closer than that, so we can't do something weird like use two synchronized probes, one sending out the RADAR pulse and another receiving it. Let's be unrealistic and suggest a 100 meter dish (whomping BIG probe!) and simplify the argument by saying the probe can only move 100 meters (the diameter of the dish) in 1.85 hours or we can't make the measurement (rotating the probe to get a 3D view makes this all worse, but let's roll with this, too...). That's a maximum speed of 54 m/s. If we use the average distance assumed for Planet 9 then we're talking about a circumference of 298.5 billion km. It'll take the probe about 175,285 years to run the orbit!
Damn you Douglas Adams!
Which means if you wanted to do this in a single human lifetime, which I'll arbitrarily define as 100 years, and ignoring the time it takes for the probes to actually enter the orbit we want, you'll need to launch 1,753 probes. Which is very likely 10 times the number of objects humanity has launched into orbit in total in human history.
My thanks to @nick012000 for pointing out that there are over 3,000 functional satellites in orbit. But I hope you'll understand my point that creating 1,753 satellites is impractical.
BUT!
What if we employ a bit of science fiction that might not be infeasible using today's tech... let's create a 1km dish using an appropriate lightweight fabric. Now the probe needs 9,465 years to run the orbit. Divide by 100 to get a lifetime and we need a scant 946 probes. That's still a honking lot of probes. You can get that down by a factor of 10 by increasing the dish diameter by a factor of 10 (and hoping that the need to rotate the probe doesn't cause its own problems. It will, but we're ignoring them for now)... but each time you do that the project becomes less feasible for different reasons (like how to store and deploy a 100 km diameter fabric dish...).
Conclusion: Not Feasible
Comment: Strong start, nice middle play, but maaan who does radar scans that way(feasibility? which feasibility? this one in the window? not talking about it) at those distances. U devide your potencial distance by a thousand as an example.  phased array antenna has let's say 10 by 10 degree transmit recievce angle, idk let's say hd resolution 1000x1000 grid/beams, and u send pulses 1000 per second, u cover the segment in a million seconds, 360 degree in little bit more than a year not a hundred thousand, eh. If purpose of sending pulse each 6 seconds in one direction is not clear, I'll explain.
Comment: But at least u tried to dig in how and will it yeild or not, not like other two, so upvote for that, it just seems u messed your calculations
Comment: @MolbOrg The radar emission can't travel faster than the speed of light in a vacuum. One billion km out, one billion back... 2x10^9/3x10^6 = 6667 seconds and ignoring rotation of the probe it can only travel 100 meters (the diameter of the dish) in that time. Ignoring all other aspects of radar (which will only make the numbers worse), divide the circumference of the orbit by the speed of the probe to get the time. What did I do wrong? The number of pulses has nothing to do with the probe speed other than to make the transit longer.
Comment: That "travel 100 meters" was another thing in u answer I'm not sure about how to interpret it. When one talk radar it means send a signal wait for reflection. U send a signal and then what? Wait for up to 12k seconds doing nothig? Cearly send signals in other directions as well, but there is a catch that arrival tme of reflection signal is uncertain and there are few ways to adress it. I guess it will greatly improve the answer if u clarify what do u mean with those meters, cuz typicaly one would think radars means rotation of antena to scan directions, why meters, cartesian strikes again, lol
Comment: @MolbOrg You're thinking about how radar works here on Earth. Let's say that you're in a plane and you're trying to detect something 1,000 miles away (that's 2/3 of the entire width of the U.S. and a LOT further than any radar is used to detect here on Earth). The entire round-trip time for the pulse is 0.01 seconds. At that speed, it's easy to spin the dish to detect things in a circle. Remember, 2 billion km. 1.85 hours round trip time. Keep the scale of space in mind and my answer will make a lot more sense.
Comment: "you'll need to launch 1,753 probes. Which is very likely 10 times the number of objects humanity has launched into orbit in total in human history." This is wrong by two orders of magnitude - there are currently over 19,000 objects in orbit around Earth according to mapping company Esri, and over 3,000 functional satellites, judging by the results of a quick Google search.
Comment: @nick012000 Objects... including all the space debris. I looked into that and discovered that the vast, vast majority of the stuff out there was not individually launched into orbit. On the other hand, you're right about Esri, which I forgot to consult. So, it's only half(ish) of the total number so far... still a whomping big number, though. Thanks for pointing it out, I'll update the answer.
Comment: I do not forget about roud-trip time, and it is exactly the reason why I wrote things in my comments, however had a mistake and should have been wrote 6k seconds instead 12k.  So it would be more helpfull if u adressed what I wrote. And as addenium, lol, current number of starlink satelites is 1558 btw, adressing u recent edit, and they plan 12k of them
Comment: Rocket launches work better near the equator because of the additional momentum from the Earth's spin. An Antarctic launch would need more fuel. How much more, I don't know, but I suspect a lot which is why countries try to launch close to the equator.
Answer: Yes, go for it, it reasonable.
Won't dig for specs of existing satelites capacities, and will go the route of JBH and handwave whatever, I hope next answers will do a better job, but I have to write this one as other answers are preoccupied with other aspect.
And current spacecrafts telescopes are quite capable at seeing stuff, so it worth take that in to account, and result can be better, but I handwave.
fleet
In a 8 inch telescope 200x magnification one can observe Neptune as blue dot from earth, and that is 30 a.u. detection range.
Assuming Planet9 orbit from 600 to 1200 a.u., we need 120 to 240 telescopes per potencial orbit plane plus minus 30a.u. which is about plus minus 3 degree inclination of potencial orbit.
Per HDE answer, and so as what I recall as well, they expect to find it in some limited section of the sky, but not certain about uncertainties, ranges of potencial plane inclinations and such, but why should it stop us let's handwave again plus minus 10 degree so we need about 3x telescopes we got for one plane
Sooo it lands us in a territory of 360-720 telescopes, let's say a thousand.
If 30kk includes full service build and launch per satelite then we have quite good(cheap), as for space program, number of 30 billions, which can bring us some result, meaning it may look as a reasonable shot to detect planet 9, it will take a long time but, for such a project it definetly cheap, from  our curent standpoint of view.
It is by a mile bigger than we do today or can do today, and in therms what it requires it looks more like Musk-Mars timeframe and capacities.
Does it make sense?
Or a telescope on the planet is a better alternative?
Telescope on the planet is not a better alternative, it a different tool, which may overlap with effort of seeking for the planet 9 efforts or do better more suitable things, but they do not cover all use cases of such probes, so telescopes no matter how many, do not render those probes useless.
In time the probes fly, they can look for asteroid objects, map kuper belt and such, magnitosphere reading as voyadger does, etc. So it an opportunity to take a 360 degree snapshot, a section of a sphere of our system of small not so spectacular but interesting things which we can't see/detect from earth. And all that just for 30 billion, "sure I take it for a dollar". So such a program makes sense it whole fligth duration.
My subjective issue for the program it a bit on a slow end, and not massive enough, but it tempting it definetly will bring interesting things.
adressing handwavium
Despite I was ready to handwave a lot, it seems not that much of handwavium here, telescope sensitivity and naked eye immediate detection of dot against possibility of detection of single photone event and smaller planet9 size and 400 timels lesser illumination seems quite balanced and within realm of current technological capacities, I would say well below our current capacities.
Signal data transmission solution are not included and there are different options, more expensive less expensive, depending on what's available, it will cost but not impossible, considering great importance and quite impressive scale and potencial for valuable data, pulling few trillions for it may create plenty of wiggle room, but if activity in space get cheaper, and maybe the program is a result(showoff case) of such efforts, then it(solutions for signal transmission) may be a sum not worth mentioning(be in billions territory), really depends on OP setting, and in Musk-Mars time may be few billions to cover it all.
Soo handwavium pretty much cancels each other, and when telescopes fail, and when a million people will be heading to Mars - it looks like a right time to launch the program, maybe a litle bit prior that, idk as part of join nasa spacex piar efforts, futile effort of nasa to rob militaries of their trillions and to test capacities of spaceships on spacex end, to launch a bunch of such probes instead of 100 red tesla bricks. That it to adress, in broad swipe, what it requires.
So feasible, makes sense.
P.S.
Pondering about it, someone needs to cook the idea and tweet it to Musk, he likes flashy stuff and it seems like another spinoff of "moderate step for a company but big step for humanity" type thing. No one done that prior, it definetly an epic feat which easily can shine on us for next 50 years.
They do have expertiese in mass sat launch and production, launch capacities, data transmissin solutions and it looks like a reasonalbe program to be funded from nasa in next 5-10 years, some initial fundings in a hundred million terrtory, cube sats for adults in some way, extending DSN, bringing it on a new level.
There is something to think about, and if there indeed is a reasonable way to connect dots between planet9, nasa interests, spacex interests, and pictures for the public, asteroid belts mapping, it may be quite a flashy project, better than starshot, which speaks that we begin solar system conquest.


Multiple robots performing a set of tasks (Optimization)

Question: Consider a situation where you have a 'm' number of robots and 'n' number of tasks. Each task is at a different location. What optimization algorithm should I use in order to assign a set of tasks to the robot. The constraints I am taking are battery percentage and distance.
How should I assign the tasks to the robots taking into account the constraints.
Answer: Try looking to Linear Integer Programming LIP where you are doing optimization by maximizing task allocation or Utility function at the same time minimizing distance and energy constraints. You will have to formulate the set of equations and use available LIP solvers.
Answer: This seems like a mTSP (multi traveling salesman problem).

Summary: The Multiple Traveling Salesman Problem (mTSP) is a generalization of the Traveling Salesman Problem (TSP) in which more than one salesman is allowed. Given a set of cities, one depot where m salesmen are located, and a cost metric, the objective of the mTSP is to determine a tour for each salesman such that the total tour cost is minimized and that each city is visited exactly once by only one salesman.


Displaying Taxonomy on the menu

Question: Friends -
Please help with this... to some of you, this is really easy. But, I have been struggling with this issue since two weeks:  I need to create categories structures such as Creating Categories/Subcategories/Subsubcategories/Products and have it appear on the menu of a newly install drupal commerce. this is what I have done so far:

I created the taxonomies necessary
I created the categories and subcategories: i.e.: Products Categories\Air Conditioners\
I created the product "Samsung Air Conditioner Unit" and inserted it into the correct categories
I clicked on All Products and observed that Samsung Air Conditioner is properly inserted into the correct subcategories
However, when I clicked on the root category "Product Category" on the main menu, which is generated by Taxonomy, I do not see the product listed.

I just need to know how to configure the content type to allow the product to display in the correct category\subcategory group.
Answer: The reason the Samsung Air Conditioner Unit was not showing as we clicked on the category was because we inserted as  Product Categories\Air Conditioner\Air Conditioning Unit instead of Product Categories\Air Conditioning Unit.
The subcategory Air Conditioner was not defined.  So, drupal did not know how to display it.  Once we moved to Product Catogoies\Air Conditioning Unit, it displayed.
Now, the task is how to create the subcategory Air Conditioners so that we could insert Air Conditioner Unit there instead of inserting it directly under Product Categories.

Displaying Taxonomy on the menu


ArcGIS 10 attribute table sort issue

Question: what is the solution for this funky sorting Esri does in attribute tables with decimal places values... screen shot: https://i.stack.imgur.com/q95Zg.png attribute table funky sort
Comment: Based on character positioning and sort results, I'd say MP is a text-type field. See answers re:character/text fields; Emily and I posted similar answers at the same time.
Answer: It looks like MP is a character field, thus the weird sort. That sort looks normal for a character field. Try creating a new field, e.g. "MP_Num", as a numeric field, then calculate the values of MP_Num based on String-to-Number conversion of MP.
edit: I just noticed the ET_STATION field looks to be the numeric equivalent of MP, so you may not need to make a new field. Try sorting on ET_STATION.
Answer: What type is your field? If it's a text type, it will alphabetise it and therefore put everything starting with 1 first. Make a new field of a numerical type and it will sort properly.
edit: Also I just tested this on a table of my own with decimal places of type DOUBLE and it does not sort like this. But it does do it for the text label field I have.
Comment: Thanks Kevin and Emily, i knew it was something simple like that... screenshot: http://i.imgur.com/E5IFv.png
Answer: Your field is a text field and if you want to sort it properly, then it needs to be padded with zeros.  There are various ways of doing this, an example is
shown here
which can be implemented within the field calculator if using ArcMap and I am sure other programs that support Python or Python-like syntax
EDIT
Then you can always use zfill to pad your rows if you are using the field calculator and the Python parser, you just .
<code>>>> "17.98".zfill(8)
'00017.98'
</code>
eg, convert to a string then pad
<code>>>> a = 17.98
>>> b = str(a).zfill(8)
>>> b
'00017.98'
>>>
</code>

ArcGIS 10 attribute table sort issue


What is the origin of the antihero Deadpool's character name?

Question: What's the story behind Deadpool's name? Most comicbook characters have names that reflect their powers, or even their costumes, but I can't make an association with either his outfit or his healing ability.
Answer: In the Deadpool and Death Annual 1999 we learn the origin of Deadpool's name; simply put, when he was being experimented on (as part of the programme which ultimately led to his powers) the guards and inmates had a dead pool where they would gamble on which subject would die next. He took his name from this.

Out of universe, ComicVine indicates that the name (and concept) were chosen as an homage to the 5th Dirty Harry movie. Although they're usually very accurate about these things, I can't find any quotes to back this up.
Comment: @Richard especially since he's a Marvel character.
Comment: The other Out-of-Universe explanation of the name I've heard has to do with him being a ripoff of Deathstroke especially in his first appearance. "Where do you do the Deathstroke? In the Deadpool."
Comment: @phantom42 - Well, indeed.
Comment: @seanduggan - good luck finding someone from DC who'll admit that...
Answer: From an out-of-continuity perspective, co-creator Rob Liefeld had a habit of using action movie titles/ and or pop culture references for character names. Hence, Deadpool, Diehard, Shaft, Bedrock, etc.
Comment: Hi. You've posted several answers already, and while we welcome your participation and enthusiasm, you need to read the [site guidelines](http://scifi.stackexchange.com/help/answering) for writing good questions and answers. There have been problems with each of your answers so far, and you should address those issues before moving on to new answers. Thanks!
Comment: No problem, I've been up all night. I'll take a more detailed run through the guidelines after I get some sleep.


Origin of Separable Verbs

Question: In what moment in the development of the German language were separable verbs introduced? Also, is there a linguistic reason behind their introduction?
Thanks!
Comment: I've addressed this question partially in [this answer](http://english.stackexchange.com/questions/25951/how-does-the-be-prefix-change-the-words-to-which-it-is-applied-how-did-it-com/26019#26019) in the English Language and Usage forum.
Comment: I learn Dutch & I know it has some similarities with German. So, is there anyone who can explain about the way of thinking and analyzing a sentence with separable verbs by a native speaker. Personally, I find it very difficult to connect the verb with its separable prefix which may stay 5-6 words further in the sentence and can be next to a preposition (which makes it difficult to choose which one is the preposition and which one is prefix of the verb). It is still ok while reading, but when people are speaking I can't understand which prefix relates to which verb.
Comment: Hi @Olga and welcome to German.Stackexchange. This answer you posted is probably more of a separate question about learning to deal with separated verbs.
Comment: I want to say two more things: English is not my native language, but I didn't get problems with pharasal verbs there. However, while I had some grammar problems with English, it helped a lot to start thinking like a native speaker. I got an enourmously interesting and unique book where 'why' questions of all grammar points were explained. It was not a grammar book, but a linguistic book.
Comment: 2) Jan, maybe it is about learning to deal with separable verbs. However, for me it is related to linguistic background. Native speakers (as well as other poeple) have limited mental resources and they can't consider the sentence of 10 words as one big whole. They break it on parts in their mind, but then you often will have two parts of a separable verb in different clusters of your memory. There should be a trick which native speakers use (without realising it). I think that knowing the history of why separable verbs are used this way can help personally me. (I would not say about others)
Comment: @Olga Welcome to StackExchange. This is not a message board / forum. It's a *question and answer site*, which means that you have to post questions *as questions* (button on the top right section) and answers *as answers*. Also, please take a look at the FAQ: http://german.stackexchange.com/faq
Comment: This discussion is interesting. I am a native English speaker who has recently started studying German and I'm trying to think about the value of separable verbs for native German speakers.
For your information: I would always say,"Turn the light off." and never say, "Turn off the light." although I have heard it said that way.
Fascinating stuff language!!
Thank you all.
Alan
Answer: Separable verbs have actually not been "introduced" but actually have always been there so to speak.
Linguists posit that separable verbs are a primitive feature of Indo European languages and can even be found in non Indo-European languages (e.g. Hungarian).
You will find separable verbs in:
A. Modern Languages

Some present-day English phrasal verbs:

To swear in: "The Prime Minister swore the whole cabinet in".
To screw up: "He is always screwing things up".

Present-day German separable verbs.

Abfahren: "Wann fährst du ab?"

Present-day Dutch separable verbs.

Schoonmaken: "Ik maakte het huis schoon" (Ich mache das Haus schön/sauber).

Even Canadian French has been "contaminated" with preposition stranding due to a prolonged contact with English (but this is a recent phenomenon, just mentioned here for the record).

B. Ancient Languages
First there is the familiar Latin sentence structure where, as in German, infinitive verbs are thrown at the end of the sentence in LIFO order.  Did Latin ever have separable verbs?  As a matter of fact "Old Latin" and a few other ancient languages did. As I read in this article of which I reproduce here the relevant part.

Many Modern [sic] Indo-European verbs are separable verbs, as in Homeric Greek, in Hittite1, in the oldest Vedic [...].

Thus, in Latin the verb supplāktum, "beg humbly", "supplicate" (adj. supplāks, "suppliant", verb plākējō, "advise, persuade"), gives sup wos plākējō (cf. O.Lat. sub uos placō), "I entreat you", and not "*wos supplakējō", as Classic Lat. "uos supplicō".

Non-personal forms, i.e. Nouns and Adjectives, form a compound (karmadharaya) with the preposition; as O.Ind.  prasādaḥ, "favour", Lat subsidium, praesidium, O.Ind. apaciti, Gk. apotisis , "reprisal", etc.

In Hittite at least the technical term is preverbs. Except for two of them, they are all separable.  The same (directional) words can be used either as adverbs, prepositions or preverbs: anda (into), appa (away), arha (outward), katta (downward), para (forward), sara (upward).  Looks familiar? Google "up" for "Hittite preverbs" for more.
Maybe the right question then could be "When and how did prepositions become [inseparable] parts of verbs?".

1 At that point, it is worth noting that Hittite is recognised as one of the oldest known Indo-European languages, if not the oldest — with such primitive characteristics as only two genders (animate/inanimate).
Comment: @user12889, the of thumb rule in English is that intransitive phrasal verbs are inseparable (as per the very definition of intransitivness: for instance "I get out") and that transitive phrasal verbs have a good chance of being separable. For instance, consider 'I look up a word in the dictionary' (*the object is **not** inserted between the verb and the preposition*), and 'I look it up in the dictionary' (*when the object is a pronoun it is **always** inserted between the verb and the preposition*).
Comment: hmm I'm not convinced by the present day English examples. One feature of separable verbs to me is that they appear non-separated in some grammatical contexts and separated in other. You would never use the verbs in the form "to upscrew" or "to inswear".
Comment: I think another example from a Germanic language might be interesting to some: **Present-day Norwegian** Phrasal verbs are abundant, but only a few separable verbs exist, like *"Jeg overtar huset"* and *"Jeg tar over huset"* both mean *"I take over the house"*.
Comment: You are probably right on an academic linguistic level. But still, as a native German speaker and long time English speaker, to me they don't quite feel like they are in the same league :-). Maybe my view is biased by the observation that German speakers don't seem to have problems with English separable verbs, but German separable verbs seem to cause a lot of pain for English speakers.
Comment: I really have to agree with user12889 basically because English present day phrasal verbs are not on the same league of separable prefix verbs that German has and I will give my 2 cents as to why. Take a look at these 2 sentances, Turn off the light.
Turn the light off. 1st one is how I always speak always always always, 2nd one is how / you can speak but for some reason i just never do, and you are right it is not like we can say offturn or onturn. So even though our phrasal verbs are technically equivalent to modern day german trennbare verben i must agree with you they are not in the same l
Comment: I don't see anybody claiming that they were the same, the answer simply says that `you will find separable verbs in [English, German, ...].`, and that `linguists posit that separable verbs are a primitive feature of Indo European languages.` The interesting thing here is that a certain phenomenon (here: separable verbs) can be found in several related languages, even though the details may have changed over time.
Answer: German has, by the way, some phrasal verbs as well in addition to the separable verbs. If you take for instance schauen (to look), you'll see that you can add prepositions like über or auf. In a sentence, these are inverted like separable verbs but are not quite the same in their infinitive form: über etwas schauen - ich schaue übers Meer. I'm not sure if this will make it easier for you to learn and understand separable verbs, but it's at least a common feature of both German and English
Comment: While German may indeed have phrasal verbs, I think your example is not the best to demonstrate that. In "übers Meer schauen", you just use the literal meaning of "schauen", nothing phrasal here as far as I can tell.
Answer: Yes, it is the writing together that distinguishes German from English, and that happens only after an evolutionary process: 'Teil/teil' was not always written together with parts of 'nehmen' and, according to Duden, 'wett' did not become consistently written together with 'machen' until the 19th century.

Origin of Separable Verbs


Linear and non-linear systems

Question: When I read about the superposition principle, it says that it works only on linear systems, my problem is that I cannot really understand the difference between a linear and a non-linear system. I cannot understand whether it means the order of the corresponding differential expression for that certain fields, or something else. So if anybody can help me get the essence of linear systems and differentiate between a non-linear one, it would be really helpful.
Comment: It's pretty much all here: https://en.wikipedia.org/wiki/Linear_system
Comment: It's basically linear algebra; it's pretty straightforward with a decent enough book.
Comment: Related, if not dupe of https://physics.stackexchange.com/q/241627/25301
Answer:
So if anybody can help me get the essence of linear systems and
  differentiate between a non linear one

Consider a simple system to be a black box with an input (stimulus) $x(t)$ and an output (response) $y(t)$.
Let $y_1(t)$ be the output given an input $x_1(t)$ and $y_2(t)$ be the output given an input $x_2(t)$.
The question is now what is the output $y_3(t)$ given the input
$$x_3(t) = a_1x_1(t) + a_2x_2(t)$$
One possibility is that the output is
$$y_3(t) = a_1y_1(t) + a_2y_2(t)$$
and then this system is a linear system (this more or less defines a linear system).  If this doesn't hold, then the system is not a linear system.
Let's work some examples to help you get the essence of this.  First, let $y(t) = Ax(t)$ and then
$$y_1(t) = Ax_1(t)$$
$$y_2(t) = Ax_2(t)$$
$$a_1y_1(t) + a_2y_2(t) = A\left(a_1x_1(t) + a_2x_2(t)\right) $$
$$y_3(t) = Ax_3(t) = A\left(a_1x_1(t) + a_2x_2(t)\right)$$
and so this is a linear system.  But, somewhat surprisingly, $y(t) = Ax(t) + B$ is not a linear system:
$$y_1(t) = Ax_1(t) + B$$
$$y_2(t) = Ax_2(t) + B$$
$$a_1y_1(t) + a_2y_2(t) = A\left(a_1x_1(t) + a_2x_2(t)\right) + (a_1 + a_2)B $$
$$y_3(t) = Ax_3(t) + B = A\left(a_1x_1(t) + a_2x_2(t)\right) + B$$
Not surprisingly, the system $y(t) = x^2(t)$ is not a linear system:
$$y_1(t) = x^2_1(t)$$
$$y_2(t) = x^2_2(t)$$
$$a_1y_1(t) + a_2y_2(t) = a_1x^2_1(t) + a_2x^2_2(t)$$
$$y_3(t) = x^2_3(t) = \left(a_1x_1(t) + a_2x_2(t)\right)^2 = a^2_1x^2_1(t) + a^2_2x^2_2(t) + 2a_1a_2x_1(t)x_2(t)$$
Finally, let's look at $y(t) = \frac{dx}{dt}$:
$$y_1(t) = \frac{dx_1}{dt}$$
$$y_2(t) = \frac{dx_2}{dt}$$
$$a_1y_1(t) + a_2y_2(t) = a_1\frac{dx_1}{dt} + a_2\frac{dx_2}{dt}$$
$$y_3(t) =  \frac{dx_3}{dt} = a_1\frac{dx_1}{dt} + a_2\frac{dx_2}{dt}$$
and so, a differentiator is a linear system.
These examples should be enough help for you to get a clear 'picture' of what the label linear system entails.
Comment: @SidharthGiri, I think I see what is confusing to you.  Note that the $x(t)$ and $y(t)$ above *do not need to be linear functions of time $t$*.  The inverse square law electric field isn't the system, it is the 'output' of the linear system (that is Maxwell's equations) to the 'input' of a point charge at the origin.
Comment: Thanks a Lot Sir, i had a question That something like the inverse square function, doesnt satisfy the condition to be linear, but for example electric fields can still be superposed..am i missing something here?
Answer: This can sound somewhat circular, but the essence of it is this:

a linear system is one that obeys the superposition system,

by definition of the former. This means that the superposition principle holds in all linear systems, but it also means that this is a relatively trivial property, and it shifts the bulk of the work into determining whether a given system is linear or not.
In more specific terms, this has nothing to do with the order of the differential operator $\mathcal L$ that enacts the system's equations of motion, but only with its linearity: i.e. we demand that
$$
\mathcal L (x+y) = \mathcal L(x) + \mathcal L(y)
$$
for all pairs of potential solutions $x,y$. This does rule out certain systems (such as e.g. $\mathcal L(x) = \ddot x(t) +x(t)^3$, for which the relation above doesn't hold), and it is in checking this linearity that the bulk of the work occurs.
Comment: @user45664 No, it's an independent postulate, but I figured it would complicate things instead of simplifying them.
Comment: Also need to have as a requirement L(ax) = aL(x).  Or maybe that follows from your single requirement.
Comment: Thanks a Lot, although i dont have much experience with operators, Can you please help Maam  , like demonstrate  how this applies to a harmonic function..?

Linear and non-linear systems


How much hash power is acutally used for dapp on the ehereum network

Question: In any given block you can see 3 eth is rewarded to the miner who found it, and
0.??? is rewarded for process transactions. Both operations require computation and storage power, but clearly finding the nounce uses 97% of the resource and process transaction use only 3% at max (gas limit * block utilization) / (3 + gas limit * block utilization)
How is dapp decentralized if one ends up host their dapp on AWS or Azure.
Answer:
In any given block you can see 3 eth is rewarded to the miner who found it,

The miner who finds the PoW is rewarded the block reward (3 ETH) and any transaction fees (and uncle rewards).

and 0.??? is rewarded for process transactions.

I'm assuming by this you mean the other miners and full nodes in the network (i.e. those who weren't successful in mining the block). If they want to run a full node, and/or continue to mine future blocks, they must verify that all transactions in the network equate to valid state changes. If they don't do this validation step, then they have no way of knowing that their node has a reliable, correct state. Any miner who can't guarantee this baseline state is going to be wasting their time (and money) mining potentially invalid subsequent blocks.
So yes, all nodes will run the transactions, but only the miner who mines the block will get paid for it.

How is dapp decentralized if one ends up host their dapp on AWS or Azure.

This is a different question. By Dapp, do you mean the front end or the back end?
The blockchain is the backend, so as long as the AWS or Azure instance that hosts your node is connecting to the main network, then it's decentralised. If it's not connected to the main network, then it's not much use for anything. (Unless you're planning to run a private network.)
If you're talking about the front end, then yes, these would usually be hosted on a centralised server. The alternative would be to use something like IPFS or Swarm.
Comment: Thanks. I am just wondering after eth switch to 100% PoS a few years from now. The 97% of hashing power used to calculated nounce then becomes free. Will the ethereum network utilize them for something else? In this case, eth automatically find some other work to perform and miner will get some sort reward by participating (not mining). Or in a difference case, miner has to choose their future payout on their own like: 1) mine other PoW coin and leave eth network; 2) provide gpu power to golem, or 3) use the gpu power to host dapp front end and/or even client side


Are all Alicorns Royal?

Question: All the Princess' in the My Little Pony, Friendship is Magic series have both wings and a horn. Is this a trend, or a coincidence? I suppose I want to know, which one implies the other? Are they royal due to their alicorn nature, or are they only alicorns because they belong to the royal blood line?
Comment: Someone's getting antsy for the next season of MLP :)
Comment: I suppose there is a direct correlation between coronation and ascension to alicorn, like, by ascending you prove yourself worthy of the crown. It's not the wings and the horn that give you the title, it's more like they are a part of your regalia; I guess an ascended alicorn could refuse the title or abdicate later, but it's the fact you ascended that proves you worthy, and the extra appendages are just a bonus. Also note all changelings have wings and horns but only one holds a royal title.
Answer: Spoiler for season 3 finale :

 Yes, though princesses are not born this way, they become an Alicorn as they become a princess.

 In the season 3 finale, Twilight Sparkle magically becomes an Alicorn.

"The others marvel at Twilight's new look, Fluttershy commenting that she looks just like a princess. Princess Celestia appears and says Twilight is a princess, explaining that since arriving in Ponyville, she's displayed the charity, compassion, devotion, integrity, optimism, and leadership that a true princess is known for."

 She is crowned the next day as Equestria's newest princess.
Answer: In general, the answer so far is "we don't know much about it" because we've only seen two three alicorns as of yet on the show. I have little doubt they'll cover the subject more in a future episode (it's one of the relative few glaring bits of history/Equestria they haven't gotten to explaining yet), but for now...
Comment: Never mind about that last sentence, i get what you're saying now (all alicorns are royal but not all royals are alicorns vs all royals are alicorns but not all alicorns are royal)
Comment: IIRC, Faust's original intent was for Celestia and Luna to be unique in the setting ("winged unicorns" at that time, not "alicorns"). However, the lore has expended since Faust's original vision, so "we don't really know" is definitely the correct answer for now.
Answer: Prince Blueblood, it may be noted, was not.  So likely no, it's just an inherent trait.  What I'd like to know is how the alicorn "gene" as it may be, passes through the royal family.  Let's break this down over-analytically:
Cadence is Celestia's niece, which means likely a third Alicorn sibling, as it's unlikely Luna ever had children.  Blueblood is not an alicorn, but part of the royal family, meaning that the gene may be by-chance recessive or dominant.  I don't if that would really work... were there, then, four siblings, three female and one male (the wing gene being dominant in one gender?)
Feel free to downvote.  That made no sense, even to me.
Comment: In _Twilight Sparkle and the Crystal Heart Spell_ (April 2013), Cadence reveals that she was born a pegasus, and was later adopted by Celestia as her niece.

Are all Alicorns Royal?


How do I unhide that Timeline event that says when I joined Facebook?

Question: I accidentally hid the date I joined Facebook from my Timeline. How can I show it again?
It doesn't appear in my Timeline to change it and make it visible. I can't see it on my Activity Log (as hidden activity). What can I do to make it appear again?
Answer: This is what was advised by the Facebook Community for How do I unhide something previously hidden? at close to the time the above was posted:

Go to your activity log on your profile and there choose "your posts". Scroll down to the post you want to 'unhide'. There will be a little barred circle next to it on the right side, click that and choose 'Allowed on timeline'. Done.

and a very little more recently (ibid.):

On your FB page look at the top right side. There is the icon for settings...Click on 'Settings">> click on "activity log">>> on the left side click on "post you have hidden">>> scroll to that post>>> on the upper right side of the post there is a circular icon>>> click on the icon and then choose the option "allowed on timeline".

However it is probably different by now (eg Posts You've Hidden is more distinct, for the moment).


Samsung Galaxy s3 mini Wifi Connected but not working

Question: I have a Samsung Galaxy S3 Mini and my Wi-Fi worked perfectly fine.
Later, I went on my phone and my Wi-Fi said it was connected but when I went on social media and stuff it said it wasn't connected. I tried everything. Please help.
Answer: First off try turning airplane mode off and on. That will often fix things. If that doesn't help, restart the phone. If that doesn't help, try hard-reset.com/samsung-galaxy-s3-mini-hard-reset.html but make sure to backup the phone data first.


Is it possible to configure a Linux server to use more than one LDAP target?

Question: We have an environment that we are tasked with managing while also giving access to projects that run the applications on the servers. Some of those projects have stood up LDAP for authenticating project users. Without getting into the discussion of why we didn't do it right in the first place we would now like to stand up our own LDAP to manage our users while not stepping on the existing LDAP configurations stood up by each project. Basically, our LDAP would be entirely separated from the project LDAP.
Is it possible to configure Linux (RHEL) to authenticate against more than one LDAP server? I'm thinking in terms of cascading authentication options: Check LDAP 1 for an account; If no account exists check LDAP 2; If no account exists check locally.
It seems like this would be possible, but having never configured LDAP I have no knowledge of whether it actually is a configurable methodology.
Answer: Yes, this is possible.

sssd with multiple domains
sssd with single domain and slapd-meta(5)
nss-ldap and pam_ldap with slapd-meta


Voicing Into Closure? [segmentation, waveform and spectrogram form Praat]

Question: I'm wondering for how long the voicing persists into closure here, if at all. At about 50% into the closure I would say it's noise only from there on, nevermind F1. But what about the oscillations before that point?
Answer: Your spectrogram is not clear. You should use a narrow-band spectrogram to see formants efficiently. So change your settings. To obtain this visualisation, you should use a small window length as it is indicated here: http://www.fon.hum.uva.nl/praat/manual/Intro_3_2__Configuring_the_spectrogram.html
After that, if it appears harmonics with a high energy, so this segment is voiced. Like in these examples: https://www.krisyu.org/blog/supp-material-cantonese-creak-perception.html
Answer: This is probably background noise. To check this, compare a section with no speech. For example, during closure of /p/ you can get very strong apparent "coarticulation" from preceding /a/ where the formants of /a/ are identifiable -- if there is an echo. You can't analyse noise with just a tiny snippet of a spectrogram, you need an actual acoustic wave sample of a full utterance.
Comment: Thank you. I am afraid that a section with no speech looks different; there is much more irregular noise. Also, there isn't much echo in the recording. And, most importantly, other word-final stop closures usually show some very obvious phonation but for about 20 m, not as in this example. For the reasons mentioned, I think it's some kind of phonation rather than just noise.


What is the state-of-the art ANN architecture for MNIST?

Question: What is actually the best neural network architecture for the classic MNIST digit classifying task? I couldn't find any that would claim to be the winner...
Answer: Actually, what I'm going to discuss is not an architecture and is like a module used in networks. It performs really well on character based data-sets like MNIST although there is other functionality for them. There is a paper called Spatial Transformer Networks written by Max Jaderberg et al. It tries to introduce an alternative for pooling layers. What it does is tries to find the canonical shape of its input by reducing transformations, like translation and rotation, or even diminishing the distortion of the inputs. It introduces a module which helps convolutional networks to really be spatial invariant. The work is amazing and the paper is easy to be read. Based on experience, using these modules may reduce the number of neurons and layers, because networks employing them won't try to learn extra stuff of the inputs, like applied transformations to the inputs. The reason is that the net will try to learn the canonical shape of inputs.

(a) shows the arbitrary input to the network (b) shows what spatial transformer has done and finally (c) is the output of the spatial transformer which can be used through other layers of the networks.
One of the significant achievements of this module is that it tries to enhance distorted inputs.

Its performance can be seen here.
Comment: Yes but this type of method will only work for images liike mnist
Comment: @Boris not really, It performs really good for tasks like face detection, or other object detections in the sceen. Take a look at the video.
Answer: As far as I know, the state-of-the-art architecture in MNIST competitions is the 2018 RMDL (Random Multimodel Deep Learning for classification) model. Check the ArXive paper.
The model is not a Network, but an ensemble of Networks, it ranked first here. Their GitHub repository is here.
Answer: You need to take a look at the state of the art (SOTA) visualisations on Papers with Code.
The answer is "A Branching and Merging Convolutional Network with Homogeneous Filter Capsules"
Answer: An ensemble of 15 CNNs classify Kaggle's MNIST digits after training on Kaggle's 42,000 images plus 25 million more images created by rotating, scaling, and shifting Kaggle's images. Learning from 25,042,000 images, this ensemble of CNNs achieves 99.75% classification accuracy. Techniques include data augmentation, nonlinear convolution layers, learnable pooling layers, ReLU activation, ensembling, bagging, decaying learning rates, dropout, batch normalization, and adam optimization. GM CDeotte explains the arch in great details here:
https://www.kaggle.com/cdeotte/25-million-images-0-99757-mnist
The SOTA link shared in the other answer uses homogeneous vector capsules generated from the final filters and gives 99.85% - a full 0.1% increase. However it is not a plain vanilla Apple to Apple conversion as the Kaggle solution involves using only a subset of the MNist images provided by the hosts. Possibly using it on the full MINST database could take it closer to the SOTA score.
The real question though is - how does this compare with the accuracy of the best SOTA CNN architecture in the universe - the human mind. While there is no formal study of the same, 99.8% was the 'claimed' human benchmark and it seems to have been broken now


Could a SR-71 be shot down today?

Question: I've read that the SR-71 had no missile defenses because it could simply outrun any adversary. Is this still true? Or is it unlikely that the SR-71 could operate unimpeded in the face of modern technology?
Comment: Umm - [yes!](http://theaviationist.com/2013/12/11/sr-71-vs-mig-31/)
Comment: Seems there is nothing new. [Mig-25 vs SR-71](http://www.wvi.com/~sr71webmaster/mig25.html). And [no newcomer in the Top 2](http://www.toptenz.net/top-10-fastest-military-airplanes.php).
Comment: Well, the SR-71 is essentially a really really big flying missile when it is in the air and we have some pretty good antimissile systems so I would take a guess and say most definitely yes.
Answer:
Could SR-71 be shot down today?

Theoretically yes. Modern day SAMs like the 40N6 that travel at speeds of Mach 6 can intercept an SR 71 within an operational range of 250 miles upto an altitude of 18 miles. Moreover, the hostile SAM will fire a salvo of missiles(at least 2-3). So the SR 71 will have to jam, drop decoys and out manoeuvre the incoming hostile missiles.Modern day SAM systems can be loaded with libraries of 3D shapes that can be used by the seeker to identify targets and threats.
S-400 and similar Russian/Chinese AD systems are meant to work simultaneously with ECM systems, meaning radars of enemy aircraft will be blinded  as 40N6 and 48N6 missiles zeroes in.
During the days of the cold war Soviet Air Force pilots were instructed to launch two rounds, a semi-active radar homing missile and a heat seeking missile.In this fashion the SR 71( or any other hostile aircraft) being targeted has a difficult problem as it must jam, decoy and/or out manoeuvre three or four tightly spaced inbound missiles
That being said getting in the general vicinity of an intercept is different from an actual intercept.
http://www.codeproject.com/Articles/19310/Hitting-a-Moving-Target-The-Missile-Guidance-Syste
From Jane's  Fighter Combat in the Jet Age : Page 116-117 "SR71 the Ultimate Target"

On June 3, 1986 an SR 71 was on a mission over the Barents Sea. Six
  Mig 31 Foxhounds, vastly superior to the Mig25 Foxbat,performed a
  coordinated intercept that would have subjected the SR71 to an
  all-angle AAM attack that even a combination of high-altitude
  maneuverability and ECM could not have defeated. Fortunately for the
  American jet, the interception took place over international waters,
  but the Soviets proved their point.

In fact,the SR 71 was retired once the USSR started deploying S 300 SAMs in large numbers.

I've read that SR-71 had no missile defenses because it could simply outrun any adversary. Is this still true?.

That is not true. The SR 71 did carry a very sophisticated Electronic Counter Measure System(ECM) called the Defensive System
One of the first MiG 31 Foxhound pilots, Captain Mikhail Myagkiy, who had scrambled with his MiG-31 several times to intercept the US super-fast spy plane, explains in his book how he was able to lock on a Blackbird on Jan. 31, 1986:

The scheme for intercepting the SR-71 was computed down to the last
  second, and the MiGs had to launch exactly 16 minutes after the
  initial alert. (…) They alerted us for an intercept at 11.00. They
  sounded the alarm with a shrill bell and then confirmed it with a
  loudspeaker. The appearance of an SR-71 was always accompanied by
  nervousness. Everyone began to talk in frenzied voices, to scurry
  about, and react to the situation with excessive emotion.

Missiles are faster and can sustain higher G loadings than SR-71.  Assuming that the SR 71 is flying at an altitude of  70,000 ft to 80,000 ft  it only needs to maneuver to out turn the missile, in order to  make the change in intercept geometry fall outside the missile's envelope. However, at extreme altitudes aircraft maneuvers are limited by stall buffet.
The SR-71 could fly a 7,000+ mile mission and the only segments of which are flown at sub sonic speeds  are the take-off, refueling  and landing.
http://www.sr-71.org/blackbird/
http://www.sr-71.org/blackbird/diagrams/index.htm
http://www.sr-71.org/blackbird/diagr...conprofile.htm
The thing is the Soviets knew that in order for any Soviet fighter aircraft to intercept an SR-71 it had to be ahead of its known flight path. These were the tactics the Soviets deployed to even get a glimpse of the SR-71. They always knew when an SR-71 launch would occur because the KC-135 fuel tanker would take off first because the SR-71 had to be immediately refueled once airborne.
Comment: Two of the links to `sr-71.org` seem to be broken.
Answer: I think that tj1000's answer is a rather romantic version.
The SR1 stopped overflying the USSR because the MIG31 Foxhound had the legs to close on the SR71 and vector it's missiles if it took off at the right moment.
Once the Soviets had locked on once and on another flight 'boxed in' the SR71 over international waters such that they had a clear kill option available, it was all over for the SR71.
The SR71 would today be toast.
Comment: As far as I know, the SR71 never overflew the USSR. That was stopped after the Gary Powers U2 shootdown. Politically, too expensive to have that happen again. The USSR did shoot down several US recon aircraft in international airspace during the cold war, all were subsonic. And the MIG31 still only has Mach 3 dash capabilities, for brief periods. Catching a single aircraft that can go Mach 3 for over an hour would take a hundred MIG31's on alert at a lot of locations, and maybe they could get it.
Comment: Even if they cannot hit it they surely can make it too busy to properly carry on it's mission (deep in enemy teritory). Considering the costs of each fly the opposition just needs to hind the SR71 long enough retiring it will be the only option left. At the bototm line my guess is survelaince sattelites can do a better (and safer) job
Answer: Certainly yes. This Wikipedia page states several missiles:

These have a greater speed than the SR-71 and a greater flight ceiling. Aso, if white house decides to turn its own enemy, then it can create a havoc with SM-3 Block 1B.
Comment: Higher speed and flight ceiling doesn't *necessarily* mean it can be shot down -- the SR-71 has a very good head start on missiles that are launched against it, meaning range is critical.
Comment: and then there's reaction time, detection range, etc. etc. Your cone for a successful intercept at those ranges and speeds is extremely small, same problem you have with intercepting ballistic missiles and even worse, the target can turn out of the cone...
Comment: Agreed with cpast and jwenting. The missile has to be so fast that it can close the distance between it and the SR-71 before it runs out of fuel.
Answer: Sadly, the SR-71 (my favorite) would be an easy target these days.
The IR heat signature from the engines and nacelles make it readily visible.
In addition to the amazing information regarding missles from the posters above,
there are distant horizon laser tracking systems and chemical lasers which can take down any aircraft (the USAF ABL system).  There are also some newer ground-based fiber lasers which can do damage to composites on both missles and aircraft at long range.
Comment: Actually, no currently operational laser system can damage an SR-71 at any relevant range/altitude. Even with perfect tracking, detection at infinite range, no clouds, and low humidity, the spot size would be too large and the intensity too low to cause any significant damage (cameras might be blinded though). Besides, unless the SR-71 flew almost directly overhead, attenuation alone would absorb most of the beam power. The [canceled] YAL only had to contend with thin-skinned, low structural-margin rockets. Even with a 1+ MW COIL, it would probably struggle against Ti designed for 300°C.
Answer: On paper, a 1980's vintage SR71 with 1980's countermeasures and support, would probably be vulnerable to the latest SAM's.
However... if a nation were to field a hypersonic aircraft with the SR71's capabilities today, they would outfit it with the latest stealth technology, support, and countermeasures. And those haven't remained static since the 1980's, either. Stealth means harder to detect (not impossible) which means shorter lead time... a real issue on a target moving that fast. Countermeasures mean more convincing decoys. Better support could be a few F35's down low, ready to pounce on any radars that go active. Maybe that's why a contemporary SR71 would be flying that mission - to get those sites to give their location away by providing a tempting target.
I suppose one could optimize a SAM to hit a hypersonic aircraft with the latest tech, but as no hypersonic military aircraft exist today, there is no reason to go to all of the expense. SAM's have been optimized to hit ICBM's, but ICBM's don't carry countermeasures or maneuver radically. They follow a fast but predictable ballistic arc.
If we take as a measure the last major threat to the SR71, it would probably be 1982, and the Libyan air strike. A SR71 flew post strike recon, when the Libyans were on full alert and mad as a nest of hornets. The Libyans threw everything they had at the SR71 and couldn't hit it, though the pilot did report 'Mach numbers he's never seen before' in completing the mission, in addition to using every ECM they had at their disposal. That was the latest US tech versus the latest Soviet tech in that era. Prior to that, a SR71 flew recon during the Yom Kippur war, and was shot at by both the Egyptians and the latest Soviet missiles, and the Israelis and the latest western missiles... as that SR71's flight hadn't been announced to them. Neither side could hit it with the tech they had.
The fact is, when the SR71 was operational, the best Soviet tech of that time couldn't stop it. Russian and Chinese missile tech has increased considerably since that time, but so have countermeasure and support tech.
Finally, the SR71 wasn't retired due to vulnerability to newer SAM systems. It was retired because the cost of operation was very high, while satellite surveillance had increased in capability at a much lower operational cost. If the USSR contributed to the demise of the SR71, it did so by imploding and reducing a major strategic threat, and the need to spend a fortune on a hypersonic aircraft to assist in balancing that threat.
Comment: You have it backwards. Hypersonic missiles (anti-ship, icbm, Sam ) will be operational and widely deployed before hypersonic aircraft.
Comment: ICBMs do carry countermeasures...
Comment: They still fly the U2, so aerial surveillance still plays an important role.
Comment: I don't think this makes sense. The whole point of the SR71 was to fly fast and high over places where you couldn't fly low, but you're proposing to accompany your "SR71-neo" by slower, lower planes, negating both of its advantages. You also suggest that the SR71 could cause defensive radars to give away their position to the lower-flying planes -- but the lower-flying planes could do that on their own, so what is the SR71 contributing?
Answer: The Lockheed A12, the predecessor to the SR71, operated by the CIA in photo reconnaissance missions over North Vietnam could sustain a maximum altitude of 95,000 ft and Mach 3.5. None were shot down, but one landed after a mission with a piece of shrapnel from an SA2 missile embedded in the underside of it's wing. In the 1950s, Surface to Air Missile systems were in their infancy, notwithstanding the development work done in Germany and Switzerland during the Second World War, and the prevailing view was still that aircraft could avoid interception by flying very high and fast, or simply very high as in the case of the Lockheed U2. This view of things was attractive at a time when rapid advances in engine and airframe technology were making possible equally rapid increases in aircraft speed and altitude capability.
Answer: The Blackbird could have shot down by the Swedes, as early as 1977.
SR-71s based in the UK would either take a route north along the Norwegian coast or east across the Baltic Sea between Sweden and Denmark.
This article is in Norwegian, so I have used Google Translate for the following two passages. The Swedish Air Force claims to have tracked and achieved "firing team contacts" on 51 occasions:
"I løpet av de 322 toktene SR-71 gjennomførte over Østersjøen mellom 1977 og 1988, gjennomførte de svenske Viggen-pilotene 51 skuddhold-kontakter ifølge Flygvapnets egne rapporter."
"During the 322 journeys SR-71 carried out over the Baltic Sea between 1977 and 1988, the Swedish Viggen pilots carried out 51 firing team contacts according to the Air Force's own reports."
=============
"Boken "Lockheed Blackbird: Beyond the Secret Missions", bekrefter i ettertid fra amerikansk hold at JA37 var det eneste flyet som klarte å komme på skuddhold for missiler. Om de hadde truffet dersom SR-71 hadde gjort unnvikende manøvre er et åpent spørsmål."
"The book Lockheed Blackbird: Beyond the Secret Missions confirms in retrospect from the US team that the JA37 was the only aircraft that managed to come on missile firing teams. If they had hit if the SR-71 had made elusive maneuvers an open question."
=============
A third passage explains that this was possible because the Swedes had better integration between Ground Control and their aircraft than the Russians:
"I motsetning til de russiske jagerflyene, var JA37 tidlig ute med en avansert taktisk computer til beregning av kurs og måldata. I tillegg kunne den ta imot informasjon på datalink direkte fra stridsledelsen på bakken. Flyradaren hadde en rekkevidde på 60 km mens det taktiske displayet rakk ut til 80 km."
"Unlike the Russian fighter aircraft, JA37 was early with an advanced tactical computer for calculating course and target data. In addition, it could receive information on data links directly from the battle leadership on the ground. The aircraft radar had a range of 60 km while the tactical display reached 80 km."

Could a SR-71 be shot down today?


Prove that $(g \circ f )^{-1} = f^{-1} \circ g^{-1}$

Question: I've already proven that if we assume f is bijective and g is bijective, then $(g \circ f)$ is bijective. I've also proven that$(g \circ f)^{-1}$ exists. I'm stuck on this part, however. Any suggestions?
Comment: Do you know the rule for groups?  The proof here is the same.
Answer: Just follow the definitions.
Suppose that $\langle x,y\rangle\in(g\circ f)^{-1}$; clearly $\langle y,x\rangle\in g\circ f$, so by the definition of composition there must be some $z$ such that $\langle y,z\rangle\in f$ and $\langle z,x\rangle\in g$. But then $\langle z,y\rangle\in f^{-1}$ and $\langle x,z\rangle\in g^{-1}$, so again by the definition of composition we have $\langle x,y\rangle\in f^{-1}\circ g^{-1}$, and hence $(g\circ f)^{-1}\subseteq f^{-1}\circ g^{-1}$. The opposite inclusion is proved similarly: all of the steps in the argument are reversible.
Alternatively, if you're not working at this low a set-theoretic level, just verify that
$$\left(f^{-1}\circ g^{-1}\right)\circ(g\circ f)$$ is the identity function on the domain of $f$, and
$$(g\circ f)\circ\left(f^{-1}\circ g^{-1}\right)$$ is the identity function on the range of $g$. These are trivial calculations if you know that composition of functions is associative.
Answer: The symbol $(g \circ f)^{-1}$ reads: the function that, when composed with $g \circ f$, gives the identity. This is a description that uniquely characterizes $(g \circ f)^{-1}$, so you only need to check that $f^{-1} \circ g^{-1}$ has this property.
Also, don't forget to check compositions on both sides!
Comment: Thank you. I had forgotten that $(g \circ f)^{-1}$ is unique. Now that I know that the solution is simple.
Answer: Recall that $(g\circ f)(a)=g(f(a))$. Three very useful statements which I will suppose you have proved before, at one point or another, are:

The composition of injective functions is injective.
The composition of surjective functions is surjective.
If $f$ is a bijection then $f^{-1}$ is a bijection and $(f^{-1}\circ f)(a)=a$.

(If you haven't seen any of these before, please leave a comment and I'll add some hints about how to prove them, but it's really an exercise of unwinding the definitions)
So we assumed that $f\colon A\to B$ and $g\colon B\to C$ are bijections, by $(1), (2)$ we have that $g\circ f$ is also a bijection, and by $(3)$ we have that $(g\circ f)^{-1}$ is a bijection as well, and $f^{-1}\circ g^{-1}$ is also a bijection. Note that $\operatorname{dom}\Big((g\circ f)^{-1}\Big)=\operatorname{dom}\Big(f^{-1}\circ g^{-1}\Big)=C$.
To show that two functions with the same domain are equal we only need to show that they do the map every element the same. Namely we need only to verify that for every $c\in C$ the following equality holds: $$(g\circ f)^{-1}(c)=(f^{-1}\circ g^{-1})(c)$$
Let $a$ denote $(g\circ f)^{-1}(c)$, then we know that $g(f(a))=c$. We also know that $(f^{-1}\circ g^{-1})(c)=f^{-1}(g^{-1}(c))$. But $g$ is a bijection therefore $g^{-1}(c)=f(a)$ from the two things we know here. But $f$ is a bijection and by $(3)$ we have that $f^{-1}(f(a))=a$, so finally we have: $$(f^{-1}\circ g^{-1})(c)=f^{-1}(g^{-1}(c))=f^{-1}(f(a))=a=(g\circ f)^{-1}(c)$$
And that what we wanted to show.


Why do books titled "Abstract Algebra" mostly deal with groups/rings/fields?

Question: As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem to 'only' cover groups, rings and fields. Why is this the case? It seems to me you'd want to study simpler  structures like semigroups, too. Especially looking at Wikipedia, there seems to be a huge zoo of different kinds of semigroups.
Comment: Well, the thing is, semigroups are *not* simpler that groups!
Comment: I am also a computer science graduate. (Although I am now determined to convert to mathematics.) It is true that semigroups are important to computer science. But I think the structures groups/rings/fields are of more importance to mathematicians. Also, if one understands the way of thinking behind these structures, it would not difficult for he or she to understand other structures, such as semigroups. For anyone, majored in computer science or mathematics or any other, abstract algebra tells people to think in the way of "structures". It is the way of thinking that is most benificial.
Answer: There is not that much substantial to say about general semigroups at an introductory level, in comparison to groups, say, and what there is to say at this level (e.g. the theory of Green's relations) tends to rely on a prior understanding of groups in any case.  (One thing I have in mind here --- but I am speaking as a non-expert --- is that semi-groups are much closer to general universal algebra than groups, and universal algebra is not, I would say, appropriate as a topic for a first course in abstract algebra.)
The theory of groups lends itself well to a first course, because the axioms are fairly simple but lead fairly quickly to non-trivial theorems, such as the Sylow theorems.  Groups (especially group actions) are also ubiquitous, and the kind of counting arguments which one can make are very useful to learn.  (It is the inability to make these sorts of counting arguments that causes so many problems when you investigate semi-groups, in comparison to the case of groups.)
The theory of fields is particularly useful in number theory, and Galois theory also provides a nice tie-in with the theory of groups, which in fact served as Galois's motivation for introducing groups in the first place. The theory of fields (especially the part to do with finding roots of polynomials) is also the part of abstract algebra that is closest to high school algebra, so it is not surprising that there is a substantial focus on it.
The theory of rings normally appears both because it is a precursor to field theory (fields are particular kinds of rings, and the polynomial rings $F[x]$ also play an important role in the study of fields), and because it includes many basic examples from mathematics, such as matrix rings, the integers, quaternions, and so on.  Rings also play an important role in the study of group representations (via the appearance of group rings), which, even if they don't appear in a first course, are just over the horizon.
Summary:
So overall, I think the answer is that groups, rings, and fields are the parts of algebra that are most closely connected to the basic core topics of mathematics, and are also closely integrated with one another.  (Groups not
immediately obviously so, but because of Galois theory and group rings,
for example.)   The theory of semigroups, by contrast, doesn't play much role in the rest of mathematics, and the theory that does exist is more complicated than the theory of groups (despite the axioms being simpler).
Comment: I would also venture to guess that groups, rings, and fields have the strongest connections outside of mathematics, for example to physics and chemistry.
Comment: I think that saying that there is "not that much substantial to say" about semigroups is rather sweeping; I would agree that there is not much to be said (or not toof ar that one can go) at an elementary and introductory level, that can rival how far you can go with groups on a first go and with little in the way of background. But there is a lot about semigroups that does not really come from groups (e.g., Isbell's Zig-Zag Lemma, which in the context of all groups reduces to a triviality). That said, I would not cover it in an intro abstract algebra course.
Comment: @Arturo: Dear Arturo, Thanks for your comment. You're correct that my statement is too sweeping, and I'll edit to better reflect the meaning (which was that is not that much substantial to say at an introductory level).  Best wishes,
Comment: You're missing a right parenthesis in the first paragraph.
Comment: This answer have a pedagogical point of view and I'm sorry if I don't trust in it. Of course the path of learning is important in book writing but there is a lot of research done with these structures, and I don't think researcher were thinking on teaching in each one.
Answer: Historically, the first "modern algebra" textbook was van der Waerden's in 1930, which followed the groups/rings/fields model (in that order).
As far as I know, the first paper with nontrivial results on semigroups was published in 1928, and the first textbook on semigroups would have to wait until the 1960s.
There is also a slight problem with the notion of "simpler". It is true that semigroups have fewer axioms than groups, and as such should be more "ubiquitous". However, the theory of semigroups is also in some sense "more complex" than the theory of groups, just as the theory of noncommutative rings is harder than that of commutative rings (even though commutative rings are "more complex" than rings because they have an extra axiom) and the structure theory of fields is simpler than that of rings (fewer ideals, for one thing). Groups have the advantage of being a good balance point between simplicity of structure and yet the ability to obtain a lot of nontrivial and powerful results with relatively little prerequisites: most 1-semester courses, even at the undergraduate level, will likely reach the Sylow theorems, a cornerstone of finite group theory. Semigroups require a lot more machinery even to state the isomorphism theorems (you need to notion of congruences).
Comment: it's clear that you don't believe it either, but the idea that one structure is "more complex" than another because it has "more axioms" is kind of silly.  It is vaguely reasonable that relational structures with more relations are more complicated: i.e., rings are more complicated than groups which are more complicated than sets.  But for structures in a fixed language, the more axioms you require, the smaller the class of models becomes.  So commutative groups are simpler than groups which are simpler than monoids which are simpler than semigroups which are simpler than magmas...
Comment: @Pete: Yes; I did use scare quotes on purpose. (-: The fewer axioms, the more models, but the less you have to "grab a hold of" when studying them (hence, 'the structure theory of fields' is easier than 'the structure theory of rings'; and we have a classification theorem for finitely generated abelian groups, but only the coclass theorems for finite $p$-groups by way of comparison).
Answer: Historical inertia.  A relatively small number of people were responsible for more or less deciding the modern abstract algebra curriculum around the beginning of the 20th century, and their ideas were so influential that their choice of topics is rarely questioned, for better or for worse.  See, for example, Section 9.7 of Reid's Undergraduate commutative algebra.
Comment: Here is a specific example.  Gian-Carlo Rota (I think it was) once wrote that it was very annoying that the typical first-year algebra curriculum does not mention tensor products, but in various other courses it is more or less taken for granted that you know what a tensor product is.  And he attributes this to historical inertia in the choice of topics.
Comment: You make it sound like some dark cabal is out to get us! :)
Comment: Well, there was already a great answer to the question "why _should_ abstract algebra books mostly deal with groups, rings, and fields" so I thought a perspective on the question exactly as it was written ("why _do_...") would be valuable.
Answer: Let me mention that my first (undergraduate!) course in algebra did start with the study of monoids.
Actually even that is not true.  It started with a categorical perspective, especially the "quotient principle": when one map factors through another.  At some point early on adjoint functors were introduced.  And then, maybe 2-3 weeks into the course, we started talking about monoids.  I especially remember the free monoid on the set $X$, which was referred to as the James construction.  A childhood friend -- who is, by one of life's little coincidences, also a mathematician; at the time he was an undergrad two years ahead of me -- came to visit me in college about a month into the quarter and he was very surprised at what he saw in the course.  Finally we started in on groups about five weeks in.  (The instructor, Arunas Liulevicius, was famous for doing this sort of thing and sort of making it work due in part to his great enthusiasm and friendly manner.)
So the concept of monoids has been with me for essentially my entire mathematical career.  I do think of this as a (slight) advantage, since a lot of people seem simply not to have this word in their vocabulary and go so far as to snicker when someone else says it.  This is ridiculous because they are a natural structure which comes up all the time, and for instance the group completion of a commutative monoid should be part of every mathematician's toolkit.
However, I agree that the general study of monoids seems not to be as rewarding as that of groups, rings, fields (or modules).  As evidence towards this, I have some lecture notes on semigroups and monoids which I wrote up a couple of years ago.  But they didn't seem to be going anywhere interesting any time soon, so I stopped at $11$ pages and may or may not get back to them later.
All in all, this may be a situation in which it is good to give the definition, a few easy examples of the definition, and then move on and allow the student an opportunity to see the structure recur in their later mathematical experience.
Comment: @MJD: It is what you say when you reveal that the answer to the question you've asked is already somewhere on the board.
Comment: On the other hand, specific types of monoids are useful in computer science and combinatorics, and monoids in categories other than Set are everywhere, e.g. monoids in Ab, monoids in Vect, monoids in Cat...
Comment: Good old Arunas. I picked up some of his lecturing techniques. "this is a test of eyesight..."
Comment: Hi, Matt!  Yes, I say that in lectures myself too, although I have never been able to muster as much glee as he had when he said it.
Comment: What does "this is a test of eyesight?" mean?
Answer: The bias against less-fashionable and/or "applied" algebra in older abstract algebra textbooks is partly a result of tradition, and  partly due to historical prejudices - not only against applied algebra but also closely related fields, e.g. combinatorics. Younger mathematicians may not be aware that, not too long ago, work in computational algebra and combinatorics got little respect from many pure mathematicians. For example, such branches of math were so shunned that leading algebraic geometers kept it secret that they were gaining insights by passing Gröbner basis calculations to the computer science department.
Thankfully nowadays most of those prejudices are old history, with a couple generations of mathematicians having been raised in the age of the computer being exposed first-hand to many diverse applications of algebra (e.g. Gröbner bases, cryptography, coding theory). Further, various authors have championed related fields that deserve respect, e.g. Rota's school of combinatorics. The pioneers knew long ago that these fields would soon come to the fore. For example, below is a pertinent excerpt from a Zbl review of the first (1974) edition of Hungerford's Algebra by a leading researcher in semigroup theory (B. M. Schein).

The book would have certainly been one of the best manuals of algebra if it
had been published fifteen or more years ago. However, nowadays it seems to
be out-of-date. What is said in the book is said clearly and well and is read
with intellectual pleasure. However, the title of the book is misleading for
no really modern book on algebra can be considered even remotely complete and
presenting an undistorted picture of algebra as it is, unless it contains
chapters on lattices and semigroups. The present book lacks both chapters.
It is patterned after the famous van der Waerden book but it disregards a
major change which occurred after the revolution begun by van der Waerden.
One cannot but agree with G. Birkhoff [Amer. Math. Monthly 80, 760-782 (1973)]
when he says: "No longer do axioms and deductive systems, patterned
after Euclid's Elements, seem so fundamental. Neither do groups or rings,
with their subgroups, subrings and morphisms... Instead, the kinds of algebraic
structures ... which are most relevant to digital computers and combinatorics
are loops, monoids and lattices (or groupoids, semigroups and semilattices),
which were largely ignored by most algebraists in 1930-1960".

If you study the history of mathematics you will soon realize that fashions come and go in mathematics just as they do in Paris or Milan. Try as we may, we are powerless to predict which little-known esoteric branch of math will soon come to the fore to play a crucial role in some application. Even Hardy's pure haven of number theory now has significant applications in cryptography. Physicists continue to marvel at the The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
So, in summary, don't let the prejudices of the past wrongly influence your studies of algebra and mathematics. Nowadays there can be no doubt that the the theory of semigroups, lattices, and combinatorics play fundamental roles in computer science (as well as mathematics).
See also below, from this deleted thread which, alas, cannot be undeleted due to a deficiency in platform design.

Speaking very roughly, the sort of "patterns" or structure that are abstracted in mathematics need not have any real-world counterparts. Let's consider a simple subject - elementary number theory. One pattern that lies at the foundation is that any nontrivial set of integers closed under subtraction has a very simple structure. Namely, it is the set of all multiples of its least positive element. Most all of the basic results of elementary number theory can be quickly derived from this simple fact (whose proof is a straightforward consequence of division with smaller remainder).
This simple pattern appears in many places, e.g. possible denominators $\,n\,$ of a fraction $\,q\,$ are also closed under subtraction $\,nq,kq\in\Bbb Z\,\Rightarrow\, (n\!-\!k)q = nq-kq\in\Bbb Z,\,$ so the least denominator divides every denominator (so it is unique). $ $ Similarly in groups, the set of $\,n\,$ such that $\,a^n = 1\,$ is closed under subtraction $\,a^n = 1 = a^k\,\Rightarrow\, a^{n-k} = a^n a^{-k}  = 1,\,$ hence the least positive such $\,n,\,$ called the order of $\,a,\,$ divides every such $\,n.\,$
This simple structure plays such a fundamental role in group theory and number theory that mathematicians have abstracted it into various forms that makes it easily recognizable, viz.  the theory of cyclic groups, and the ideal theory of Euclidean  domains, whose ideals are all principal, i.e. the set of all multiple of the "smallest" element.
There are also many common divisibility patterns, e.g. the universal properties of gcd and lcm
$$\begin{align}
& a\mid b,c \iff a\mid \gcd(b,c)\\
& b,c\mid a \iff {\rm lcm}(b,c)\mid a \end{align}\qquad$$
Thus whenever ones sees the pattern on the LHS, it may be replaced by the pattern on the RHS. This, combined with the basic laws of gcd and lcm,  goes a long way towards solving many divisibility problems, e.g. see various posts here on gcds and on lcms.
In mathematics we strive to abstract out these fundamental patterns in their most general form. Doing so maximizes the chance that they will be easily recognizable in diverse contexts, and hence can be efficiently reused. Though the recognition and abstraction of these non-real-world patterns is probably not something that evolution has trained us for, one can become quite proficient at such. There have been studies of such symbolic reprogramming of our innate mental faculties, e.g. see the classic study Thought and choice in chess by the psychologist Adriaan de Groot. Just as grandmasters have trained their minds to efficiently recognize the many fundamental abstract patterns of pieces that arise in (human) chess games, so too have mathematicians trained their minds to recognize fundamental abstract mathematical patterns.
Answer: Most books on abstract algebra that I've seen tend to start with or want their students to have some sort of number-theoretic background.  Lattices, semilattices, Boolean Algebras, De Morgan Algebras, and the like probably have gotten ignored for this reason.  Heck, I'd believe some people would consider all of those structures as part of logic, not mathematics (and there there exist disparaging comments about Lattice Theory from people like G. H. Hardy).  Also, semigroups don't fit well, since most of the common number systems have more structure than that.  The positive natural numbers form a commutative semigroup, but you don't see that in many texts either.
You hit the nail on the head about a huge zoo of semigroups.  I tried to do some thinking on this on my own after reading Pete's link, and instead of talking about examples of semigroups, it seemed better to talk about classes of examples of semigroups.  With some infinite semigroups I could form subsemigroups on the fly by remembering ordering properties of numbers, and then deleting the lower (or higher) end members (e. g. ([0, 1], $\ast$) is a semigroup with $\ast$ as multiplication, so then delete some interval of real numbers [x, 1] from [0, 1] and since multiplication always decreases, closure holds, and I think that implies association comes for free for such a subsemigroup (see the link).  You can't get examples this easily with groups (and I doubt with rings or fields), so perhaps finding examples of groups, rings, and fields seems more impressive.
Additionally, the Schaum's Outline of Group Theory introduces groupoids, then semigroups, then monoids (though it doesn't use that name), and homomorphisms, isomorphisms get expressed in terms of a groupoid first, and Cayley's theorem get discussed for semigroups first also.
Answer: Semigroups and monoids are at the heart of algebra.  In fact, the majority of algebra is really monoid theory in different guises.  Groups are but monoids whose element are invertible.  Rings are monoids along with an "addition" operation (another monoid!) that comes along for the ride.  Modules are the actions of such monoids.  Algebras are monoids of modules.  So, it is quite perplexing to me that the standard algebra text books don't lay the proper conceptual foundations for the student, which are surely in monoid theory.  One rare book stands out as a shining exception:
Claude Chevalley: Fundamental Concepts of Algebra, Academic Press, 1957.
The moral is that if any one reads algebra with semigroup/monoid goggles, they will find them pretty much everywhere.  For my own personal education and enjoyment, I have been collecting Notes on Semigroups as I read the traditional algebra, which I am happy to make available for general reading.  (But, please note that these are written as personal notes for myself and they will change and expand as I go along.)
However, note that there are plenty of standard text books available on semigroups.  Some widely used ones are:
Clifford and Preston, The algebraic theory of semigroups, AMS, 1961.
Howie, J. M, An introduction to semigroup theory, Academic Press, 1976.
Higgins, Techniques of Semigroup Theory, Oxford Science Publications, 1992
There are also some very nice algebraic automata theory books that deal with semigroups in considerable detail:
Eilenberg, S., Automata, languages and machines, Academic Press, 1976
Holcombe, W.M.L., Algebraic automata theory, Cambridge University Press, 1982
Pin, J.-E., Mathematical foundations of automata theory, online book for now.
Answer: Any theorem on groups is simultaneously a theorem of monoids, semigroups and magmas, since each layer is defined as a refinement of the previous layer. Once you finish a book/course in abstact algebra and move on to applying it in your field, there is a plethora of resources for using the theory of semigroups and monoids in computer science, so you'll be set.
Comment: Well, the cancellation laws is a theorem on groups, but it not true for monoids or semigroups.
Comment: Oh right, I had it backwards! A theorem on magmas is a theorem about semigroups, which is a theorem about monoids, which is a theorem about groups. Still though, a course in group theory is useful when one later encounters a CS book that uses semigroups/monoids a lot. Anyway, thanks for clarifying my sloppy thinking!


Style input type=number

Question: I have this input type:

input[type=number] {
    width:9vw;
 height:10vw;
 max-width:42px;
 max-height:52px;
 text-align: center;
    -moz-border-radius: 1px;
    -webkit-border-radius: 1px;
 border-radius:1px;
 text-align:center;
 border:3px solid black;
 display:inline-block;
 margin-left:6%;
 background-color:black;
 color:white;
}
<input type="number"  MIN="0" MAX="15"  default="2" >

I noticed that the display changes with different browsers. I I would set the arrows always visible and which cover the entire height. I would only white arrows on a black background. I need also to set default value to 2. thanks for your help
Comment: Do you want to set the arrow in the input box as well????
Answer: You don't use default.  You use value:
<input type="number"  MIN="0" MAX="15"  value="2" >

Your min and max will be ignored.  That should be handled with Javascript.
That is not true. It actually works fine.  Plus, it even auto-filters bad values.
As far as the looking different part...  You have a condition in your style to allow for different browsers.  That may or may not be overridden at render time.
Comment: Part of this is incorrect; the `min` and `max` values are pretty well adhered to as long as set in an HTML5 document. http://www.w3schools.com/tags/att_input_min.asp
Answer: As durbnpoisn said, you can simply set a value attribute to fix your default value problem.
Since you gave your input field a type, you're referencing to the HTML5 input field, which automatically includes those arrows. To get some custom arrows, you have to play around with -webkit-appearance: none;. In order to do so, I've found 2 simple explanations to achieve this here and here.
Combined together this would look somehow like this:
<input type="number" value="2">

input[type=number] {
  width: 9vw;
  height: 10vw;
  max-width: 42px;
  max-height: 52px;
  text-align: center;
  -moz-border-radius: 1px;
  -webkit-border-radius: 1px;
  border-radius: 1px;
  text-align: center;
  border: 3px solid black;
  display: inline-block;
  margin-left: 6%;
  background-color: black;
  color: white;
}

input::-webkit-outer-spin-button,
input::-webkit-inner-spin-button {
    /* display: none; <- Crashes Chrome on hover */
    -webkit-appearance: none;
    margin: 0; /* <-- Apparently some margin are still there even though it's hidden */
    background: #eee url('https://i.stack.imgur.com/YYySO.png') no-repeat 50% 50%;
    width: 14px;
    height: 14px;
    padding: 4px;
    position: relative;
    right: 4px;
    border-radius: 28px;
}

You can also check this in a little fiddle.

Style input type=number


how do I know which node picks up my transaction? in general, how is a node assigned to broadcast a specific transaction?

Question: How do I know what node is broadcasting my transaction to the network? how can i get more information about this node? for example, how do i know if the node is a geth or parity node? does it matter?
Answer: You're missing the blockchain concept.
Transactions are processed on all nodes. Mining determines the order of the transactions all nodes eventually agree on. This is not the same thing as a node running transactions and the rest of the network accepting its findings.
My answer over here may shed some more light on this. how blockchain can handle the concurrency?
Hope it helps.
Comment: this is very helpful, thank you. i am wondering, though, about the pre-accepted transactions. to restate my question more clearly, there has got to be a node that picks up my transaction first, correct? Is this the node closest to me? when i send my transaction to the network through metamask, where does it physically go? do the popular nodes (metamask, mycrypto) have known physical locations?
Comment: Several details to unpack. Generally, your request goes to your own node which then broadcasts to others that you _want_ to do something. Getting picked up in a block by a miner is the critical step to _have done_ something. MetaMask uses the Infura backend which is a web service with a known endpoint. Infura is an alternative to running your own node.