Wikipedia:Reference desk/Archives/Science/2012 January 4

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January 4

Is caring for sources a kind of bias?

If I dismiss some sources (like Yahoo Answers and the like) when searching for information, is that a kind of bias or even fallacy (if what I've seen is rubbish, then the rest must be too)? On the other hand, it seems difficult to accept all the sources as potentially valid...88.8.76.174 (talk) 00:43, 4 January 2012 (UTC)

Claiming "all information on Yahoo Answers is useless" is biased. Some of it is correct. Claiming "it is not possible to know if an answer on Yahoo Answers is correct or not because the interface does not require the users to provide verified credentials or even a valid reference" is not a bias. It is a description of Yahoo Answers. -- kainaw™ 00:54, 4 January 2012 (UTC)

Yes, it is bias and discrimination, though based on experience, and therefore justified. It is also a logical fallacy, as you point out. See genetic fallacy. However, estimating the probability of a source providing useful information is an indispensible element of research, especially when there is an abundance of sources to be evaluated. In this case, one's time is (more probably) better spent looking elsewhere than at sources which have proven unreliable in the past. Doing so is logical, based on an empirical evaluation of probability, and the fact that we don't have infinite time to evaluate all of the available sources. Dominus Vobisdu (talk) 01:06, 4 January 2012 (UTC)

Note that while one expert source trumps one random guy on the internet, one expert source may not be as accurate as many random guys on the internet. For example, if you asked an expert the atomic weight of tin, you'd probably get the right answer, but he could make a mistake, such as a typo. If you asked 100 random people on the internet, some answers would be dead wrong, but the majority, or at very least the plurality, would be right. Now this is a trivial example, but the same applies to more complex questions, too. This theory is why many think that stock prices reflect the value of a company so well. So, if you reject the consensus of the public in favor of a single expert, then, yes, that's an unwarranted bias. StuRat (talk) 01:46, 4 January 2012 (UTC)

It's true that there's problems with sourcing, and potential biases introduced through the sources used. For starters there's often a reporting bias, meaning certain things are more likely to be reported on by what we regard as reliable sources than other things. Then supposed reliable sources are not always as reliable as we may hope. For example I almost invariably find that when something significant enough happens in my local area to be reported in a major news source, almost inevitably there will be some (usually) minor detail or details wrong - misspelled name, inaccurate distances, other minor errors that any local would know to be wrong that while not overly significant are still wrong, heck, even the times I've been 'quoted' in the media almost invariably they have tweaked what I 'said' to suit themselves; not enough to totally change my meaning, but often enough to be not something I'd actually say. Now if this information was turned into an article on Wikipedia, those blatantly incorrect factoids could be reffed as being true and at times would be virtually unchallengeable. On a similar note I have seen a website I have created and another website a friend created cited as sources on WP; not that they are incorrect, but it's interesting if I made something up and added it straight to WP it could likely be challenged, but if I make it up, put it on another site, then that site gets cited that's regarded as sourced. Nonetheless, giving supposedly reliable sources is about the best we can do at this stage, but also why we allow many unsourced claims to remain in articles. After all, sometimes that 'random guy on the internet' does know better than a 'reliable source'. --jjron (talk) 12:05, 4 January 2012 (UTC)

Epistemologically speaking, one random guy could trump an expert, and probably does on a regular basis. There is nothing that says being a certified expert makes one always correct. And there is nothing that says the crowd should necessarily be correct, either. (As plenty of "scientific literacy" studies have shown, going to large numbers of people actually doesn't help on many questions.)

The reason I do not generally trust Yahoo Answers is because my prior experiences with it show it not to be correct most of the time. That is itself a fallacy — it is the problem of induction — but I don't let that bother me much. I do not endeavor to be perfectly logical; rough heuristics work fine most of the time. --Mr.98 (talk) 12:33, 4 January 2012 (UTC)

If only you theorised that Yahoo! Answers is unhelpful most of the time, for reasons, then you'd be avoiding induction just fine.  Card Zero  (talk) 09:51, 5 January 2012 (UTC)

Caring about bias is a form of bias; metabias. Von Restorff (talk) 14:10, 4 January 2012 (UTC)

how to calculate the tolerance of parallel resistors

for example,R1(10k,1%),R2(20K,5%),if the two resistors are parallel,what's the total tolerance of them? — Preceding unsigned comment added by 121.15.144.10 (talk) 02:04, 4 January 2012 (UTC)

If you compute R = 1/( 1/R1 + 1/R2 ), first using both resistors' values at their extreme low end, and then again at their extreme high end, what do you get? -- ToE 03:32, 4 January 2012 (UTC)

I don't think there's an easy way to do this. As the above suggests, you have to work it out the long way. However I don't think you need to work out the low AND high, you just work out the expected value and compare it to either the low OR the high. i.e. I believe a 10K 1% resistor falls between 9.9K and 10.1K, not 9.5 to 10.5, that would be 0.5%. In the example, the expected value is 6.66K, work out the "low" using 9.9K and 19K and get the percentage of the difference to get the circuit tolerance. Vespine (talk) 04:06, 4 January 2012 (UTC)

Actually, scratch that, maybe you do need to work out the low and the high.. I don't have the time right now to confirm, it doesn't seem to affect your example, but I'm not sure if there are circumstances where it would make a difference. Vespine (talk) 04:11, 4 January 2012 (UTC)

The low and high limits will indeed be off by different percentages. Using the highest possible values for both resistors gives 6.81994 kΩ, which is 2.299% higher than the nominal parallel resistance. But using the lowest possible values for both resistors gives 6.50865 kΩ, which is 2.370% lower than the nominal parallel resistance. Red Act (talk) 04:47, 4 January 2012 (UTC)

Our article propagation of uncertainty addresses this sort of question from a more rigorous statistical point of view, but is almost certainly well beyond the scope of what is being asked here. -- ToE 05:11, 4 January 2012 (UTC)

You may also want to read Engineering tolerance and Tolerance analysis. Red Act (talk) 05:25, 4 January 2012 (UTC)

Very good. From Tolerance analysis (which is written from a mechanical engineering point of view): "Arithmetic tolerance stackups use the worst-case maximum or minimum values of dimensions and tolerances to calculate the maximum and minimum distance (clearance or interference) between two features or parts." -- ToE 06:27, 4 January 2012 (UTC)

"Lowest possible" and "highest possible" are incorrect. There is no absolute cutoff for the variation of the resistances, just a statistical probability that they will exceed the stated variation. A larger variation is possible some percentage of the time. If you had 100 identical 10% resistors in parallel, it would be incorrect and misleading to state that the "largest possible" resistance would be calculated by assuming each was 10% higher than nominal. Take statistics into account. Edison (talk) 14:18, 4 January 2012 (UTC)

The right way to do that is by performing a partial differentiation. if R is a function of R1 and R2 $R=R(R_{1},R_{2})\,$, than the uncertainty $\Delta R\,$ is given (to first order) by $\Delta R={\frac {\partial R}{\partial R_{1}}}\Delta R_{1}+{\frac {\partial R}{\partial R_{1}}}\Delta R_{2}$. -- Dauto (talk) 16:10, 4 January 2012 (UTC).

I believe there is a minor typo in the above. Dauto, shouldn't the formula be:

$\Delta R={\frac {\partial R}{\partial R_{1}}}\Delta R_{1}+{\frac {\partial R}{\partial R_{2}}}\Delta R_{2}$

...in other words, the formula for the total derivative of R with respect to R1, R2. Nimur (talk) 16:40, 4 January 2012 (UTC)

For those interested in comparing numbers, with R = (R₁^-1 + R₂^-1)^-1, Dauto's first order approximation would be ΔR = ( R₂ / (R₁ + R₂) )² ΔR₁ + ( R₁ / (R₁ + R₂) )² ΔR₂, which for this case is (4ΔR₁ + ΔR₂)/9, yielding a tolerance of ±2.3333%, which is very close to the average (2.3346%) of the upper (2.2990%) and lower (2.3702%) tolerances computed directly. For series resistors with R = R₁ + R₂, the first order approximation will, of course, be equal to the exact values. Note that the use of the total derivative simply offers a quick method of approximating the value, and does not address the statistical concerns expressed by Edison and rebutted below. -- ToE 01:01, 6 January 2012 (UTC)

I don't quite understand why Edison says it's incorrect and misleading to calculate using the high or low values. Real world resistors don't follow a perfect bell curve distribution. The tolerance IS an absolute cutoff for the variation of the resistance. If you have a resistor that falls outside of its tolerance, it is faulty and you chuck it out. In fact, I'm pretty sure some 1% resistors are just the ones which tested within 1% of their intended value picked from the processing line which makes regular 5% tolerance resistors. Batches of resistors can also easily be skewed in a direction, so their tolerance won't necessarily statistically "even out". Manufacturing processes these days are very good and you'll probably find that most 5% resistors you get are more like 3% resistors, but "in principle" there's no reason why you could get a whole batch of 100ohm 5% resistors which measure 95 ohms. Vespine (talk) 22:20, 4 January 2012 (UTC)

Kindly provide a reliable source to support your assertion that all 5% resistors absolutely and positively fall within 5% of nominal, rather than the 5% tolerance being related to a statistical distribution, with some percentage probabilistically falling outside the 5% tolerance. Edison (talk) 05:49, 5 January 2012 (UTC)

Riedon, a resistor manufacturer[1], defines "tolerance" as "Max. allowed deviation of the absolute resistor value from the nominal resistor value in percent from the nominal resistor value. (delivery tolerance)".[2] Page vii of the book Tolerance Design of Electronic Circuits says "Thus, the actual value of a 10% tolerance 10kΩ resistor may be anywhere between 9kΩ and 11kΩ."[3] Red Act (talk) 16:01, 5 January 2012 (UTC)

Well this may be a case where the process changed from many years ago, before computerized testing of each resistor became feasible. It appears that it is now common to test every resistor, rather than doing statistical tolerancing, [4], of a portion of a batch of resistors, which could have allowed a few outliers,beyond the stated tolerance limits, since the only requirement was the sample standard deviation being small enough that 2 sigma or 3 sigma would be within the tolerance band. Edison (talk) 19:30, 6 January 2012 (UTC)

Maybe many years ago, but computerized? I doubt it, seems trivial to make resistors fly through a test circuit for final "tolerance testing" without a computer. Vespine (talk) 22:20, 8 January 2012 (UTC)

What it boils down to is: if you're building a circuit (or a bridge, or anything), and you determine that a tolerance or error of X % is absolutely required in order for Feature Y to work ... you'd better make sure you know what your supplier means when they say "X % tolerance." The difficulty is defining "absolutely", "requires," and "%" in context. This is why engineers are paid a lot more than 8th-graders, even though a surprising quantity (95%, ±5%) of real-world engineering mathematics could be done by the eighth-grader. Nimur (talk) 22:55, 4 January 2012 (UTC)

'Many years ago' (maybe still) a common practice was to make the resistors then measure them and sort them into bins of different nominal values. This resulted in a nearly flat distribution of values across the tolerance band. Manufactures may even select-out those closest to the nominal value for sale as more expensive high-precision components. Taking away the 'good' resistors creates a gap in the distribution of the values of the cheap low-precision ones - they're guaranteed to differ from their nominal values (and I have refs. for this: Tim Williams, The circuit designer's companion and Nihal Kularatna Electronic circuit design: from concept to implementation). --catslash (talk) 23:42, 8 January 2012 (UTC)

Why was Stjerneborg subterranean?

Our article Stjerneborg mentions that it was subterranean, just like this picture here. But why was that necessary? The Danish article (Google translation) talks about protection from wind gusts. Obviously, you wouldn't want to hang your quadrant on a wooden wall that shakes in every wind, but if you use masonry, you should be stable enough to not have to worry about wind. Conversely, the big problem for Tycho, atmospheric refraction, is increased. The closer to the ground you are, the more ground effects you will get - in the extreme case, you might even get a mirage. BTW, thanks to Mr.98, who made me first aware of Stjerneborg here. — Sebastian 06:53, 4 January 2012 (UTC)

That Google translate result gives a plausible explanation: "This new observatory was built underground, and only during observations was left observation hatches open. In this way we got a better accuracy, and then also better manage temperature fluctuations helped to better obervationer." Remember, in the 1500s there were no thermostats! So an underground chamber would be an effective way to keep a more constant temperature. Wnt (talk) 08:55, 4 January 2012 (UTC)

Interesting idea. Let's look at the effect of temperature on the instruments. What effects are there? I can think of thermal expansion. The coefficient for brass is about 20 10⁻⁶/K, or 200 μm/m for ΔT=10K. This may be in the order of magnitude of a very good naked eye. However, for measurements of angles, linear expansion is irrelevant. Unless the structural parts of an instrument are made of vastly different materials for the different directions, that effect seems to be negligible. Am I overlooking anything? — Sebastian 10:24, 4 January 2012 (UTC)

Actually, Tycho's eyesight must have been even better than that; he claimed that 5″ were easy to discern, which is 25 μm/m.

Read this. Our article Tycho Brahe says: "Tycho Brahe was granted an estate on the island of Hven and the funding to build the Uraniborg, an early research institute, where he built large astronomical instruments and took many careful measurements, and later Stjerneborg, underground, when he discovered that his instruments in the former were not sufficiently steady.". More info can be found here. On tychobrahe.com it says: To further improve the accuracy of the position measurements, Tycho found that it was necessary to shield the instruments from disturbing wind gusts, and to have the instruments on solid fundaments. This led him to design a new observatory in 1584, Stjerneborg. Located partly underground, and with hatches to open in the observation direction only, a new level of accuracy was obtained. At the same time temperature fluctuations were suppressed, further improving the accuracy. His instruments were not just made of brass, and for example wood can be influenced by temperature as well. See Wood warping. Von Restorff (talk) 11:10, 4 January 2012 (UTC)

One other thought: Hven was probably a very cold place at the end of the 16th century (which was right at the beginning of the Little Ice Age). The underground arrangements may have also been for the been for the benefit of himself and his assistants, making observations through long northern nights. --Mr.98 (talk) 12:26, 4 January 2012 (UTC)

The OP mentioned vibration from the wind, and I reckon this is quite close to the mark. If you have a strongly magnifying telescope, the last thing you want is for it to wobble around every time a horse and cart goes past outside. Henry Cavendish had a similar solution (i.e. an underground laboratory) when he was 'weighing the world'. Unfortunately, I've been unable to find any sources that give details about his house, which I believe had rather a labyrinth of labs and workshops underneath. I don't think I made it up, though. - Cucumber Mike (talk) 21:08, 4 January 2012 (UTC)

While Is suspect wind was a factor, I will just point out that Brahe had no telescopes whatsoever, as they had not yet been invented. This is all pre-magnification technology (like the sextant shown above). --Mr.98 (talk) 22:38, 4 January 2012 (UTC)

Not to mention the mural quadrant (the linked article shows Tycho himself using one of his). Such instruments, which had also been built by other cultures, were more accurate the bigger they were (assuming they were precisely constructed). Tycho's assistant and successor Johannes Kepler was able to demonstrate that the orbit of Mars (and by inference those of the other planets) is an ellipse (not a circle as had previously been assumed) by using the naked eye positional measurements made with Tycho's non-telescopic instruments. During Kepler's lifetime the astronomical telescope was invented (he himself made early improvements to it) and thereafter incorporated into such astrometrical devices. {The poster formerly known as 87.81.230.195} 90.193.78.46 (talk) 23:21, 4 January 2012 (UTC)

Tycho Brahe's sextant, used for measuring the angular distances between stars. This instrument has a wooden frame.

Thanks everyone for your replies so far. This is getting more and more interesting as we move along. Above all, I'm excited about the link to Astronomiæ instauratæ Mechanica; that's beautiful! I didn't know that existed. And you're right, Von Restorff, at least some of the instruments did use wood, such as the one you pictured here. At first I thought: 'Well, if they used wood, no wonder they had problems.' But then I realized, much to my surprise, that wood is actually more stable than brass. Douglas-fir parallel to grain, e.g., has a thermal coefficient of expansion of 3.5 (our article, orig. source), which is 1/6 of that of brass. But it comes even better: The dreaded warping may not have been a problem. According to this US Forest Service document, "Wood is dimensionally stable when the moisture content is greater than the fiber saturation point." Their statement may not have been meant with the precision needed for astronomical measurements, but at least Tycho seemed to agree: "The other parts are only wooden, but they have been combined and joined together in such an ingenious way - we have described above when we explained the armillae - that they cannot bend or change with changes of the air, and furthermore the whole is covered with painted canvas."*

Either way, we are now talking about effects of the same order of magnitude as the precision of the naked eye. So we need to look at the arguments for climatic stability. Let's start with hygric (moisture) stability. Moisture fluctuations will be interior (people, fire) or exterior. The former won't change by moving underground, and as long as you have a gaping hole in your wall/ceiling, the latter will be mostly due to simple convection through that hole, the size of which doesn't change by moving underground. Conversely covering the instruments with canvas, as they did, will have a stabilizing effect. (I presume it was pointed with oil to make it less permeable.) Thermic (temperature) stability will be improved by going under ground, but, if you compare it with solid stone masonry, which already has a high thermal mass, the effect is less than that of burning more wood, which is incidentally how Europeans in general reacted to the little ice age. (I at least haven't heard of people moving their castles underground to save energy then.) I apologize for just claiming "it's less" without actually doing the math; I'm a bit in a hurry now. I just wanted to post this already to firstly thank you for the good discussion so far, and to encourage people to add more ideas. — Sebastian 02:13, 5 January 2012 (UTC)

I assume he was smart enough to think about the long term effects of the weather on his beloved instruments - even over periods of hundreds of years - and he understood the importance of guarding them against any outside influences, however tiny they may be. Wind speed increases with increasing height above the ground, starting from zero due to the no-slip condition.

This instrument with its small crypt-turret is covered on top by a roof, made of small, smooth beams, ingeniously joined together and connected, below the horizontal top of the wall and outside the azimuth circle, by a strong, round wooden ring. Hidden inside this ring are wheels, placed opposite each other in four places. With the aid of these wheels the roof can be turned around, with little effort, as may be desired. In this way the two oblong windows, which are placed in the roof opposite each other, and which are likewise formed of small beams, can be turned towards any star that is to be observed. These windows are shut when the observations are finished, in order that the instrument should be protected against damage from the air, and against rain and wind. This should be done, as far as possible, with the other instruments also, since they are very valuable and are easily damaged if not protected against violence as well as dirt and the influence of the elements.

On the upper part of the wall is a carefully constructed roof that protects the instrument against damage by weather. This roof also has its windows, which can be opened, and it can be turned around as required by the observations on wheels that carry it. Furthermore, the windows can be shut while the instrument is adjusted, in order that the wind should not force its plumb-lines D out of their natural position. When the instrument is not used, they protect it against damage by rain and wind.

At 06:53 you wrote: "you wouldn't want to hang your quadrant on a wooden wall that shakes in every wind, but if you use masonry, you should be stable enough to not have to worry about wind"; but this quote above is about an instrument made from steel supported by a strong iron pillar, while deep down in the ground it rests upon a stone foundation.

Von Restorff (talk) 03:58, 5 January 2012 (UTC)

Placing scientific buildings partially underground in order to increase stability is not confined to the past - I used to work in a Metrology lab which was cut back into a hillside in order that the precision weighing balances could be mounted on concrete plinths bonded directly to the bedrock and isolated from the rest of the building (vibrational instability is a major cause of measurement uncertainty in precision weighing - we used to achieve smaller uncertainties than labs with better balances simply because we were on granite and several miles from the nearest road with regular traffic). I can therefore confirm that the strategy mentioned above works, and is a very good idea if the precision of your instruments is high enough to warrant it. Equisetum (talk | contributions) 10:53, 5 January 2012 (UTC)

I've been fortunate to afford some time to read Tycho's description of Uraniburg (page 76ff) and Stiernburg(page 83f), and I'm surprised that he, despite otherwise going into much detail about his choices, has very little on this question. His only direct statement on height is this: "First of all, the place should be in a high locality from where there is a free view round the whole horizon [...]" (page 76). He does mention his purpose for it being subterranean: "My purpose was partly to have placed some of the most important instruments securely and firmly in order that they should not be exposed to the disturbing influence of the wind, and should be easier to use [...]". The first point is what Equisetum said above, which also fits to what the German article says: "The observatory was built mostly underground, so that the instruments rested beneath the sand layers on solid ground." - Google does a great job translating this.) However, that isn't completely convincing. I still think that building a solid, vaulted masonry basement would have provided practically the same stability while minimizing the surface effects.

So, I have a new idea. Tycho was an experimenter, after all. He probably wanted to know how much wind contributed to the exactness of his measurements. What better way to find out than build another observatory at a different height above ground? In the continuation of the above quote he writes (anticipating the idea behind double-blind trials!): "... partly to separate my collaborators when there were several with me at the same time, and have some of them make observations in the castle itself, others in these cellars, in order that they should not get in the way of each other or compare their observations before I wanted this." Unfortunately, I haven't seen his results; I'm not sure if he published them. Maybe they were just inconclusive, or he didn't do the comparisons systematically enough to feel safe to publish them.

With this, I now realize that I had been misled by a statement in our article that "he found Uraniborg [not] stable [...] enough for his precision instruments", for which, as I now realize, there is no basis. Rather to the contrary, he once writes (p. 78) "I was to erect buildings which with their solidity and magnificent equipment were suitable for astronomical work". Therefore, I will change that misleading statement, and I think that closes the matter. Thanks everybody for your help! — Sebastian 01:32, 6 January 2012 (UTC)

Yvw. It may be a good idea to ask the museum if they can give you the email address of an expert on the subject; lots of experts are glad to share their knowledge. Von Restorff (talk) 03:12, 6 January 2012 (UTC)

Actually, I did, albeit through the webpage tychobrahe.com. They just haven't replied yet. By now, I have reached closure, so I'm not waiting as impatiently for a reply. If I did, I would certainly also try the link you provided, which are the Danes who provided the wonderful online book. — Sebastian 07:00, 6 January 2012 (UTC)

White blood cells

Are white blood cells present in the blood capillary near the lungs? — Preceding unsigned comment added by Yacopo (talk • contribs) 07:48, 4 January 2012 (UTC)

Yes. This paper discusses leukocytes (a type of white blood cell) in lung capillaries. SmartSE (talk) 12:50, 4 January 2012 (UTC)

Yes; for the most part the constituents of the blood don't change in different parts of the body, even though some other things do, mainly gas concentrations. Or are we misunderstanding your question? --jjron (talk) 15:18, 4 January 2012 (UTC)

All (or perhaps nearly all) alveolar macrophages were once upon a time monocytes within the lumina of capillaries within the lung parenchyma. Macrophages are extravasated monocytes. DRosenbach ^{(Talk | Contribs)} 05:17, 5 January 2012 (UTC)

precession

Where's the axis?

there's something I don't understand in gyroscopic precession.the magnitude of the torque is said to be Mgr.well, when calculating that, they look at the rotation of the wheel around a different axis than that of its initial rotation, L.so when they say torque=dL/dt, it's a different torque, I know I'm misleaded, but how? — Preceding unsigned comment added by Irrational number (talk • contribs) 11:02, 4 January 2012 (UTC)

Could you please cite a source for this? Who spoke or wrote about gyroscopes, and left you perplexed? I recall the equations and their explication as being very basic and intelligible, even to one like me who had difficulties with higher mathematics. Edison (talk) 05:43, 5 January 2012 (UTC)

Perhaps the Earth's orbital precession constitutes a gyroscope. Take a look at moment of inertia, which describes the calculations for different shapes. Perhaps you're saying that when the axis of rotation precesses in an angle such that the axis is no longer perpendicular to the wheel, that the formula would thereby deviate from its original state? ~AH1 ^(discuss!) 18:56, 5 January 2012 (UTC)

well, the last section of the 12th chapter of "principle of physics".the axis for which the equation of torque is written,is the one about which the precession is occurring, but the one for which L is written for, is the axis about which the rotation of the wheel is occurring.but I kinda solved the problem in my mind by saying "both the torque and the angular momentum shown in the Picture can fully describe how the wheel moves." but I still feel a little confused...--Irrational number (talk) 21:33, 5 January 2012 (UTC)

We'll skip the formulas here

i'm not sure whether the humanities or science desk is better for this: I'm looking for examples of science texts where the author says let's skip the maths, and moves onto the conclusion - this is research for a piece of fiction, so the field doesn't matter -- what I'm hoping for is just some samples of the language used in such cases, especially in vintage texts -- the older the better. Thanks very much

Fermat's last theorem may fit, or else it's close. Grandiose (me, talk, contribs) 11:08, 4 January 2012 (UTC)

What level of works are you looking for? There's a lot of texts written for lay audiences that do this, as well introductory books pointing out the implications of a more advanced result; or just pointing out that such and such advanced result is needed, but outside the scope of the current volume. Research level stuff will skip a lot of elementary stuff and get to it's conclusions, which is a little different than the former (hence, my question). :-) Phoenixia1177 (talk) 11:22, 4 January 2012 (UTC)

Dr Riemann's Zeros by Karl Sabbagh is another one, which gives a fascinating insight into one particular theorem and the process behind its proof. --TammyMoet (talk) 12:46, 4 January 2012 (UTC)

A common phrase used in math texts is "details are omitted". A google books search for /"details are omitted" math/ will give you thousands of hits, such as "The proofs are completely analogous to those of the two preceding theorems and the details are omitted." SemanticMantis (talk) 14:36, 4 January 2012 (UTC)

..."Left as a exercise for the reader...." I'll have to dig to find a specific text that used the phrase, but it and its variants were the bane of many of my classmates. Nimur (talk) 16:13, 4 January 2012 (UTC)

Agree. But that is mostly textbook language. In a paper, I'd expect something like "the proof is strictly analogous to X" (where X may be "theorem one" or "the argument on page 3"). --Stephan Schulz (talk) 16:16, 4 January 2012 (UTC)

Einstein gave popular lectures in multiple continents in the 1920's. I doubt that a fraction of a percent of the audience could have understood any mathematical notation beyond basic arithmetic. Did he describe special and general relativity in purely verbal "layman's language?" Edison (talk) 05:39, 5 January 2012 (UTC)

Thanks heaps, all.'Details are omitted' and 'Left as an exercise...' are exactly the sort of thing I'm looking for - especially the latter, so double thanks for that - and yes, they'll be very helpful search terms. Adambrowne666 (talk) 06:54, 5 January 2012 (UTC)

Another popular phrase sometimes used to simplify proofs, or avoid tedious detail, is Without loss of generality. AndrewWTaylor (talk) 09:32, 5 January 2012 (UTC)

That's just a reduction in the detail of a proof by use of symmetry — not quite the same thing. Another phrase used is "the proof is outside the scope of this text" (or similar). Dbfirs 09:49, 5 January 2012 (UTC)

Agree about Without loss of generality, which is similar to "let X be arbitrary, but fixed". What I also have experienced is "After some tedious calculations, we arrive at..." (note that if you use that, you better get those right ;-). --Stephan Schulz (talk) 10:42, 5 January 2012 (UTC)

You might try viewing video interviews with Richard Feynman, such as this. Not a text, but the transcript of such may be considered one. ~AH1 ^(discuss!) 18:33, 5 January 2012 (UTC)

Wattles in goats

What is the purpose of wattles in mammals? Kittybrewster ☎ 15:57, 4 January 2012 (UTC)

They serve no purpose (or we simply don't know the purpose) and are believed to be a genetic trait "left over" from evolution in goats and pigs. There is an explanation of the genetics of wattles in goats. Von Restorff (talk) 16:13, 4 January 2012 (UTC)

Apparently in pigs and sheep the wattle gene does have an effect on overall growth. [5] [6] I didn't quickly find information I could access about what chromosome the wattle gene maps to - if we had this for sheep, goats, and pigs, I wonder if we could "map" the gene simply by statistical analysis of which orthologs are on the right chromosome in all three species. But it's also possible they're non-homologous genes. Also note that just because a mutation interferes with wattle formation doesn't mean you have to have the wattle to avoid the effects of the mutation. Looking at them I'm reminded of the wing and haltere arrangement in Drosophila wings, but I can't really picture that applies - more likely, based on the propensity of humans to play with earrings, there's some kind of sexual selection going on? But I don't know. Wnt (talk) 03:10, 5 January 2012 (UTC)

The sexual selection theory seems likely, but in some populations the wattle gene is at the brink of extinction. Von Restorff (talk) 05:36, 5 January 2012 (UTC)

It's erectile tissue. ~AH1 ^(discuss!) 18:29, 5 January 2012 (UTC)

ice cap and tundra

Is ice cap climates basically high elevation or low elevation. Is it like highland plateau or is it inland valley low elevations like Verkhoyansk. Is inland part of Greenland or Antarctica is just far inland like Yakutsk, Russia. Is tundra allowed to have trees at all. most people's understnading is tundra is a frozen swampland where trees cannot grow at all. I am serious about there is actually trees growing in Qaqortoq, Greenland. Is Qaqortoq, Greenland tundra or it can't be tundra because of the green trees really existing there. Does Kangerlussuaq have permafrost or no permafrost. Kangerlussuaq summer high actually exceeds 60 F continously at summer I though Kangerlussuaq is more closer to ice sheet than Nuuk. --69.229.39.25 (talk) 22:28, 4 January 2012 (UTC)

I'm not sure I fully understand your questions: I'll try to address them in the order you ask, but it would make things easier in future if you ask fewer questions at one time and separate them more clearly. Please refer to the articles I've linked, because they mostly contain fuller information relevant to your questions.

• The Greenland and Antarctic ice caps (strictly they are ice sheets, larger than ice caps) typically have high elevations, because the ice itself is mostly quite thick, up to several kilometers in their interiors, though lower at coastlines where the glaciers meet the sea.
• There are generally no ice-free lower elevations resembling the areas around Verkhoyansk in an ice cap's interior; the ice is so thick that only mountains are high enough to rise through it, and they are mostly bare rock because of the elevation and low temperatures.
• Inland Greenland and Antarctica do not resemble inland Siberian areas with soil and trees like Yakutsk, they are under kilometers-thick ice.
• There are different kinds of tundra in different parts of the world (you can click on the World map in that article to enlarge it and see which parts of Greenland have tundra). Most tundra has some bushes and other vegetation, but it has no trees, which is what the word "tundra" actually means: however, in some places it has a few scattered and small trees dwarfed by the cold climate. Note that although tundra is frozen in Winter, its surface layers above its permafrost do thaw in Summer, which is why plants can grow there.
• Qaqortoq is in the extreme south of Greenland and does not have permafrost (see the section on Climate in that article) and therefore is not on tundra.
• Kangerlussuaq is futher north in Greenland and is surrounded by areas of tundra (with underlying permafrost) but the town itself is on low land at the head of a fjord, where the warmth of the sea may prevent year-round permafrost from persisting. Most Arctic and Antarctic areas can get quite warm in Summer because for nearly six months the sun is up 24 hours per day, so there is no cooler night-time as there is in the the Temperate zones.

I hope the above answers your questions: feel free to ask more if there is still something you aren't clear about. {The poster formerly known as 87.81.2301.95} 90.193.78.46 (talk) 05:06, 5 January 2012 (UTC)

Trees that do grow in permafrost can often be hundreds of years old, as the roots take time to penetrate through the frozen ground and aquifers, especially when the ice is near the surface or very thick. I'm not sure whether permafrost is equal to tundra, but note that often the frozen ground is tens of meters under the surface, or may be discontinuous (non-year-round) permafrost. When that permafrost melt, the trees often become drunk. ~AH1 ^(discuss!) 18:26, 5 January 2012 (UTC)

Polonium - to - actinium gap

Why are all the elements from polonium to actinium extremely rare and unstable compared to the elements both before (such as lead and bismuth) and after (such as thorium and uranium) this gap? Whoop whoop pull up ^{Bitching Betty | Averted crashes} 23:52, 4 January 2012 (UTC)

See Magic number (physics). Dauto (talk) 00:44, 5 January 2012 (UTC)